LEARNING GUIDE

MATHEMATICS DBE

MTH-4271-2
MTH-4253-2 SCI
VOLUME 1
ALGEBRAIC AND
GRAPHICAL MODELLING
IN A FUNDAMENTAL CONTEXT 1

E
I ANC W
MPLEN E
WIT
CO
I N H TH
G R A M
P R O S T U DY
OF
RÉSOLUTION LEARNING GUIDE
MATHEMATICS DBE

MTH-4271-2 SCI
VOLUME 1
ALGEBRAIC AND
GRAPHICAL MODELLING
IN A FUNDAMENTAL CONTEXT 1
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 Table of Contents VOLUME 1
How the Learning Guide is Structured . . . . . . . . . . . . . . . . . V

CHAPTER 1 CHAPTER 2
Interpreting Numerical Information . . . . . . . . . . . . . . . 2 Revealing the Secrets of Paranormal Activity . . . . . . 64
Step Functions and Greatest Integer Functions Algebraic Expressions

SITUATION 1.1 SITUATION 2.1

STEP FUNCTIONS RATIONAL EXPRESSIONS
GREATEST INTEGER FUNCTION f (x ) 5 [x ] PERFECT SQUARE TRINOMIALS
GREATEST INTEGER FUNCTION f (x ) 5 a [bx ] DIFFERENCE OF SQUARES
SP 1.1 – Pay-As-You-Go Calls. . . . . . . . . . . . . . . . . . . . . . . . . . . 4 SP 2.1 – The Prodigy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Exploration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Exploration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67



Acquisition A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Acquisition A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
• Interpreting a Step Function • Recognizing Rational Expressions
• Determining the Rule for a Greatest Integer Function • Determining Restrictions
from its Graph • Adding and Subtracting Rational Expressions
• Determining a Missing Value Algebraically Using the • Factoring Perfect Square Trinomials
Rule of a Greatest Integer Function
• Simplifying Algebraic Fractions Through Factorization
Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Acquisition B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Acquisition B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
• Analyzing the Effect of Reversing the Signs of Parameters a and b.
• Recognizing a Difference of Two Squares
Consolidation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 • Factoring Using the Difference of Two Squares

Consolidation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
SITUATION 1.2
GREATEST INTEGER FUNCTION f (x ) 5 a [b (x 2 h)] 1 k SITUATION 2.2
NEAREST INTEGER FUNCTIONS MULTIPLYING TWO POLYNOMIALS
SP 1.2 – Measuring the Temperature. . . . . . . . . . . . . . . . . . . 26 DIVIDING A POLYNOMIAL BY A BINOMIAL
Exploration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
SP 2.2 – The Telepath. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Acquisition A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Exploration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
• Discovering Greatest Integer Functions of the Form
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Acquisition A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
f(x) 5 a[b(x 2 h)] 1 k
• Graphing Greatest Integer Functions of the Form • Multiplying Polynomials
f(x) 5 a[b(x 2 h)] 1 k • Dividing a Polynomial by a Binomial Without a Remainder
• Determining the Rule for a Transformed Greatest
Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Integer Function Using its Graph
• Studying Nearest Integer Functions Acquisition B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
• Dividing a Polynomial by a Binomial With a Remainder

Acquisition B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Consolidation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

• Writing a Greatest Integer Function from a Table of Values

• Interpreting the Properties of Greatest Integer Function KNOWLEDGE SUMMARY. . . . . . . . . . . . . . . . . . . . . . . 104
f(x) 5 a[b (x 2 h)] 1 k
INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Consolidation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

KNOWLEDGE SUMMARY. . . . . . . . . . . . . . . . . . . . . . . 50

INTEGRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

LES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
ANSWER KEY PAGE XXX III
 TABLE OF
CONTENTS

COMPLEMENTS

REFRESHER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

KNOWLEDGE SUMMARY. . . . . . . . . . . . . . . . . . 121

MATHEMATICAL REFERENCE . . . . . . . . . . 131

GLOSSARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

ANSWER KEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

EVALUATION GRID . . . . . . . . . . . . . . . . . . . . . . . . . . 181

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IV TABLE OF CONTENTS
 TABLE OF
CONTENTS

HOW THE LEARNING GUIDE

IS STRUCTURED
Welcome to the learning guide for Algebraic and Graphical Modelling in a Fundamental Context 1.
The aim of this course, which is the first in the Secondary IV Science sequence, is to develop
your ability to handle situations that require an algebraic or graphical model to express a relationship
between quantities. To this end, you will learn about three new real functions:
• step functions;
• greatest integer functions;
• second degree polynomial functions.

You will complete your learning by expanding your knowledge of:

• the properties of straight lines on a Cartesian plane;
• solving systems of equations;
• algebraic manipulations.
You will be required to use various solution strategies to understand and model situational problems.
You will need to use your mathematical reasoning skills. You will also have to describe how you solved
these problems clearly and thoroughly using mathematical language.
You are now invited to complete the learning activities found in the three chapters of this guide and
enrich your knowledge of algebra.

Portailsofad.com
Go to portailsofad.com for videos,
ICT activities and printable versions of
resources that are complementary to the
SOLUTIONS series, which you can use
throughout your learning journey.
© SOFAD / All Rights Reserved.

V
 TABLE OF
CONTENTS

CHAPTER COMPONENTS
The learning process followed in each chapter enables students to progress by building on what they
have learned from one section to the next. The following diagram illustrates this approach and specifies
the pedagogical intent of each section.

CHAPTER INTRODUCTION
The first page describes the context and theme that
will serve as a backdrop for the acquisition of the
new knowledge discussed in the chapter.

CHAPTER 1

Step Functions and Greatest

CHAPTER 1
Integer Functions SITUATION 1.1 A table of contents
STEP FUNCTIONS
accompanies this first

SITUATION 1.1
EXPLORATION
GREATEST INTEGER FUNCTION f(x) 5 [x]

Interpreting Numerical GREATEST INTEGER FUNCTION f(x) 5 a[bx]

SP 1.1 – Pay-as-You-Go Calls p. 4 page. The knowledge
Information
SITUATION 1.2 to be acquired is
E
ach day, we receive a phenomenal amount of information GREATEST INTEGER FUNCTION f(x) 5 a[b(x 2 h)] 1 k
in numerical form. Whether on our phones, computers,
tablets or televisions, we are constantly bombarded with
NEAREST INTEGER FUNCTIONS
SP 1.2 – Measuring the Temperature p. 26
described for each of
numbers. Often, these numbers are rounded off to make them
easier to express. Just think about cellphone calls billed to the
KNOWLEDGE SUMMARY p. 50
the Situations, as well
next minute or the temperature outside rounded off to the
nearest unit. The same is true for race times, which are rounded
down to the nearest tenth of a second. These rounded values
INTEGRATION p. 56 as the theme of the
LES
can be modelled using functions called step functions. In this
chapter, you will learn about this type of function, including how
Timekeeping p. 62 situational problems.
to describe them and how to represent them using both graphs
and algebra.
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© SOFAD / All Rights Reserved.

2 CHAPTER 1 – Step Functions and Greatest Integer Functions 3

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STEP FUNCTIONS
GREATEST INTEGER FUNCTION f(x) 5 [x]
SITUATION 1.1 GREATEST INTEGER FUNCTION f(x) 5 a[bx]

Pay-as-You-Go CallsSITUATION 1.2 SP 1.1

GREATEST INTEGER FUNCTION f(x) 5 a[b(x 2 h)] 1 k
NEAREST INTEGER FUNCTIONS

Annie, who is studying at

university far from home, has a
Measuring the Temperature SP 1.2

cellphone that she rarely uses.

That is why she prefers to pay for
each call she makes based on its
length instead of getting a fixed- A weather website shows the current
price monthly plan. temperature in a few towns in the
Laurentians and the surrounding
Annie knows that the number of minutes for areas. All of Québec’s weather stations
each call on her bill has been rounded off. measure
November temperature
bill precisely, to the
For example, on November 30, she tried to hundredth of a degree. However, tem-
Type of Length
get in touch with a friend who was not there.
Date peratures
Number are
call usually
(min) broadcast to the
Cost ($)

SITUATIONS
Although this call lasted barely 15 seconds, November 3 public as whole
819-256-XXX3 Out numbers.
4 1.60
1
or __ of a minute, she was billed for a full November 5 450-231-XXX1 Out 2 0.80
4
minute. Her contract with the phone November 10 450-231-XXX1 In 1 0.40

company is clear in this regard: “Each call November 10 450-231-XXX1 Out 5 2.00

is charged at a rate of $0.40 per minute; November 15 514-634-XXX8 Out 1 0.40

In general, there are two any fraction thereof is rounded up to the November 22 819-321-XXX0 In 3 1.20

nearest minute.” November 23 450-652-XXX8 Out 16 6.40

November 30 514-354-XXX2 Out 1 0.40
Since Annie does not want to pay for time

learning Situations per chapter.

November 30 514-354-XXX2 Out 8 3.20
she did not use, she decides to switch Total 41 16.40
cellphone companies. A new company on
© SOFAD / All Rights Reserved.

the market is offering the following rate in

The approach taken in these its contract: “Each call is charged at a rate of
$0.20 per complete block of 30 seconds.” Therefore, if Annie makes a call of less than 30 seconds, there will
be no charge—which seems like a very appealing offer.

situations makes it possible to

These numbers are automatically rounded off by data processing software, without any human intervention.
Annie has a monthly budget of $30.00 for phone calls. She makes about 10 calls per month to catch up with her
© SOFAD / All Rights Reserved.

All temperatures measured across Québec are thus processed in a fraction of a second, using a real function.
friends and family— 2 or 3 calls a week—so she wants to know what the average length of each call should be.
It is important to understand how temperatures are rounded off in order to interpret them properly. For

acquire new knowledge and

example, as you certainly know, the freezing point of water is 0 °C; if the temperature rises above this point,
Using the rule describing the situation, calculate how longsnow
Annie’s
andcalls should
ice melt. Forbethe
onpeople
averagewho run the ski resort at Mont-Tremblant, it is important to know the real
TASK

so she does not go over her monthly budget with the newtemperature
phone company.so theySupport
can planyour
which slopes should be open or shut and inform skiers. In this learning
calculations with a table of values or a graph. situation, you need to provide Mont-Tremblant’s operators with a mathematical tool for making sound

develop mathematical skills in decisions based on the temperature.

real, realistic or purely

You must determine the function rule for rounding a number to the nearest integer so that the
TASK

Mont-Tremblant ski resort can automate its information process. You must also determine the
interval containing the real temperature at Mont-Tremblant when the weather report says it is
4 CHAPTER 1 – Step Functions and Greatest Integer Functions 0 °C outside. Show your work and support your answer with graphs.

mathematical contexts.
26 CHAPTER 1 – Step Functions and Greatest Integer Functions

VI HOW THE GUIDE IS STRUCTURED

 TABLE OF
CONTENTS

PHASES OF EACH SITUATION

STEP FUNCTIONS
GREATEST INTEGER FUNCTION f(x) 5 [x] Mathematical knowledge
SITUATION 1.1 GREATEST INTEGER FUNCTION f(x) 5 a[bx] EXPLORATION ACQUISITION A targeted:
• interpreting a step
function

Pay-as-You-Go Calls
• determining the rule for
SP 1.1
Answer the questions in this exploration activity to review the mathematical terms related to functions and 1. Step Functions a greatest integer
identify the characteristics of the function you can use to model Situational Problem 1.1. You will find them function from its graph
In order to round off numbers of any value, it is important
useful when building your graph and determining the function rule.
to study a type of function called step functions.
• determining a missing
value algebraically, using
the rule of a greatest
1 Explain the differences between the two billing rates. A step function is a discontinuous function that is constant

SITUATION 1.1

SITUATION 1.1
integer function

ACQUISITION A
EXPLORATION
over certain intervals called steps.
Annie, who is studying at
university far from home, has a 1 Consider the following graph of a step function.
cellphone that she rarely uses.
That is why she prefers to pay for
2 To make sure you clearly understand the situation, calculate the cost of a 100-s call on each rate. y
Explain your calculations step by step. 5
each call she makes based on its 4
length instead of getting a fixed- 3

price monthly plan. 2

1

8 27 26 25 24 3 22 1 1 2 3 4 5 6 7 x
Annie knows that the number of minutes for
2 2 2

2
2

each call on her bill has been rounded off. November bill 3
2

4
For example, on November 30, she tried to Type of Length
2

Date Number Cost ($) 5

2
get in touch with a friend who was not there. call (min)
Although this call lasted barely 15 seconds, November 3 819-256-XXX3 Out 4 1.60 3 Apply the procedure you described in your previous answer to calculate the cost of the following calls.
1
or __ of a minute, she was billed for a full November 5 450-231-XXX1 Out 2 0.80 a) On which intervals of x is each step in this step function?
4 a) A 200-s call
minute. Her contract with the phone November 10 450-231-XXX1 In 1 0.40

company is clear in this regard: “Each call November 10 450-231-XXX1 Out 5 2.00
b) What are the domain and the range of the function illustrated above?
is charged at a rate of $0.40 per minute; November 15 514-634-XXX8 Out 1 0.40

any fraction thereof is rounded up to the November 22 819-321-XXX0 In 3 1.20 Domain: Range:
nearest minute.” November 23 450-652-XXX8 Out 16 6.40 b) A 300-s call
November 30 514-354-XXX2 Out 1 0.40
Since Annie does not want to pay for time November 30 514-354-XXX2 Out 8 3.20 REFRESHER EXERCISES
she did not use, she decides to switch Total 41 16.40 REMINDER PAGES 115 TO 117, QUESTIONS
4 TO 9
cellphone companies. A new company on

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© SOFAD / All Rights Reserved.

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the market is offering the following rate in Representing Intervals
its contract: “Each call is charged at a rate of STRATEGY Give yourself numerical examples
An interval is a set consisting of all the real numbers found between two endpoints. It may be
$0.20 per complete block of 30 seconds.” Therefore, if Annie makes a call of less than 30 seconds, there will When you want to analyze a function situation, it is helpful to give yourself numerical examples. For instance, to represented in words, in a graph, in set-builder notation or in interval notation.
be no charge—which seems like a very appealing offer. determine the rule for a function, you can often simply replace the numbers in the examples with variables.
Annie has a monthly budget of $30.00 for phone calls. She makes about 10 calls per month to catch up with her Example:
friends and family— 2 or 3 calls a week—so she wants to know what the average length of each call should be. It is clear that the new company offers a better deal on calls. You probably also noticed that the cost remains The interval of the numbers from 2 (included) to 5 (excluded) may be represented
constant for a certain interval of time and then increases by $0.20. • graphically:
2 5 R
Using the rule describing the situation, calculate how long Annie’s calls should be on average
4
TASK

How long will a call last before the cost increases by $0.20? • in set-builder notation: {x  R | 2  x , 5}
so she does not go over her monthly budget with the new phone company. Support your
calculations with a table of values or a graph.
• in interval notation: [2, 5[

4 CHAPTER 1 – Step Functions and Greatest Integer Functions ANSWER KEY PAGE 145 5 ANSWER KEY PAGE 145 7

SITUATIONAL PROBLEM EXPLORATION ACQUISITION A

Linked to the main theme of the This section invites you to analyze This is where the knowledge
chapter, this page briefly describes the data of a situational problem, needed to solve the situational
the context of the situational and then to identify the knowledge problem is assimilated. Each
problem, as well as the information that you possess and the Acquisition encourages reflection
required to solve it. knowledge you need to acquire before presenting new
A box describes the task you will in order to perform the task. mathematical knowledge.
have to perform later in the Solution The questions posed will guide you
section. This task is the starting toward a problem-solving strategy.
point for acquiring new knowledge
to solve the situational problem.

Mathematical knowledge
SOLUTION ACQUISITION B targeted: CONSOLIDATION
• analyzing the effect of
reversing the signs of
parameters a and b
You can now solve Situational Problem 1.1.
SITUATION 1.1
STEP FUNCTIONS
GREATEST INTEGER FUNCTION f(x) 5 [x] 1. The Effect of Reversing the 1 Let the function f(x) 5 6[0.25x].
Signs of Parameters a and b
GREATEST INTEGER FUNCTION f(x) 5 a[bx]

Pay-as-You-Go Calls SP 1.1

a) Evaluate the following expressions.
In the task about cellphone contracts, billing practices were one of the main differences between the two 1) f(8) 5
Annie, who is studying at
university far from home, has a companies: one company billed by rounding up to the next integer, while the other charged by the integer
cellphone that she rarely uses.
2) f(23) 5
That is why she prefers to pay for
each call she makes based on its but did not count incomplete blocks of time. This type of difference changes the appearance of the graph as
Using the rule describing the situation, calculate length instead of getting a fixed-
price monthly plan.
well as the rule of the greatest integer function. 3) f(5.8) 5
how long Annie’s calls should be on average so Annie knows that the number of minutes for
TASK

each call on her bill has been rounded off. November bill

she does not go over her monthly budget with the

For example, on November 30, she tried to
get in touch with a friend who was not there.
Although this call lasted barely 15 seconds,
Date

November 3
Number

819-256-XXX3
Type of
call
Out
Length
(min)
4
Cost ($)

1.60
To visualize the effect of reversing the sign of parameter a or b, a graph of the function f(x) 5 30.5x is plotted 4) f(p) 5
on a Cartesian plane below. The accompanying questions will help you understand how the graph reflects
1
or __ of a minute, she was billed for a full November 5 450-231-XXX1 Out 2 0.80

new phone company. Support your calculations

4

b) What is the range of f?

minute. Her contract with the phone November 10 450-231-XXX1 In 1 0.40

company is clear in this regard: “Each call November 10 450-231-XXX1 Out 5 2.00

the changes in the parameters.

is charged at a rate of $0.40 per minute; November 15 514-634-XXX8 Out 1 0.40

with a table of values or a graph. any fraction thereof is rounded up to the

nearest minute.”
November 22
November 23
November 30
819-321-XXX0
450-652-XXX8
514-354-XXX2
In
Out
Out
3
16
1
1.20
6.40
0.40
Since Annie does not want to pay for time November 30 514-354-XXX2 Out 8 3.20
she did not use, she decides to switch Total 41 16.40

1.1 Reversing the Sign of Parameter a

cellphone companies. A new company on
© SOFAD / All Rights Reserved.

the market is offering the following rate in

its contract: “Each call is charged at a rate of
$0.20 per complete block of 30 seconds.” Therefore, if Annie makes a call of less than 30 seconds, there will
c) What are the critical values of f?
be no charge—which seems like a very appealing offer.
Annie has a monthly budget of $30.00 for phone calls. She makes about 10 calls per month to catch up with her

1
friends and family— 2 or 3 calls a week—so she wants to know what the average length of each call should be.

Using the rule describing the situation, calculate how long Annie’s calls should be on average
If parameter a becomes negative in the initial function f(x) 5 30.5x, the function rule becomes
TASK

so she does not go over her monthly budget with the new phone company. Support your
calculations with a table of values or a graph. f1(x) 5 230.5x. d) Graph the function.

a) Evaluate the following expressions.

4 CHAPTER 1 – Step Functions and Greatest Integer Functions 2 Let the function f(x) 5 210[20.1x].
SITUATIONAL PROBLEM FROM PAGE 4 1) f1(23) 5
Summary of the Facts a) Evaluate the following expressions.
2) f1(22) 5 Tip 1) f(212) 5 2) f(26) 5
• Annie has a monthly budget of $30.00. 3) f1(21) 5 By studying the critical values of a greatest integer
b) Is the function increasing or decreasing? Justify your answer.
• On average, she makes 10 calls a month. 4) f1(0) 5 function, it is easier to get an overall picture of the
function and, consequently, to extract information on
• A new company is offering a contract that reads as follows: 5) f1(1) 5 the impact of the parameters.
c) Based on the values of parameters a and b in the function rule,
“Each call is charged at a rate of $0.20 per complete block of 30 seconds.”
6) f1(2) 5 identify the following.

Representation of the Situation b) Plot the function f1(x) on the same 1) The direction of the steps in the graph
y
(Table of values or graph) Cartesian plane as function f(x).
8
f(x)
© SOFAD / All Rights Reserved.

© SOFAD / All Rights Reserved.

© SOFAD / All Rights Reserved.

c) What type of reflection did the graph 2) The length of the steps
of function f1(x) undergo when the sign 6
of parameter a was reversed?
4 3) The height of the jumps

2
d) Is this an increasing function or a
decreasing function? d) Graph the function.
28 26 2 4 2
2 2 4 6 8 x

2
2

4
2

6
2

8
2

16 CHAPTER 1 – Step Functions and Greatest Integer Functions 18 CHAPTER 1 – Step Functions and Greatest Integer Functions 22 CHAPTER 1 – Step Functions and Greatest Integer Functions
© SOFAD / All Rights Reserved.

SOLUTION ACQUISITION B CONSOLIDATION

By the time you reach this section, In this second acquisition, you will This section will allow you to
you should have acquired all the acquire new knowledge prescribed consolidate the mathematical
knowledge and strategies that are by the program linked to the knowledge acquired in Acquisitions
essential to solving the situational knowledge encountered in A and B. Like the Integration section,
problem described at the Acquisition A. Consolidation also helps with the
beginning of the situation. development of mathematical skills.

AT THE END OF A CHAPTER...

Voici un résumé de tous
KNOWLEDGE SUMMARY intEgration
les savoirs À RETENIR.
Écrivez les information
LES
s
manquantes.
This section
La fonction summarizes
définie par parties all the In 1thisUnesection,
usine de bacswhich includes
de recyclage exercises
en restructuration décide d’engager une The LESdirectrice
nouvelle is a complex
des ventes task developed
qui aura pour mandat d’augmenter les profits de l’entreprise au cours de la prochaine année.
knowledge to Remember in the form
Une fonction définie par parties, c’est une fonction dont la règle diffère and complex
selon l’intervalle situations, you will have to according to the certification evaluation
À l’arrivée de la nouvelle directrice, le profit de l’entreprise représentait une perte de 6000 $. Les deux
dans lequel se situe la variable .
of fill-in-the-blank questions. We invite applypremiers
the knowledge
mois à la direction seen in this
lui ont permis chapter.
de ramener model.
le profit à une valeur It islesaccompanied
nulle. Pour deux mois by a
suivants, l’entreprise s’est mise à faire des profits de 1000 $ par mois. Puis, pendant deux autres mois,
youExemple
to fill: in the missing information. les profits ont suivi la règle f(x) 5 0,5x 1 4, où x est exprimé en mois et competency evaluation grid.
f(x) en milliers de dollars. 2

Représentation graphique Finalement, pendant les six derniers mois de l’année, les profits ont augmenté à un rythme équivalent
Règle de la fonction
au taux de diminution des profits des deux mois précédant cette période.
VII
IRS EN RÉSUMÉ

y
apitre 1
ApITRE 1

tégration

Représentez graphiquement cette situation en considérant que les changements entre les différentes
4
périodes de temps se font de façon constante.
3
0,5x 2 1 si 0  x  8
 TABLE OF
CONTENTS

COMPLEMENTS

SELF-EVALUATION REFRESHER
This last activity will prepare you for the final exam of the course and will help you to determine your level of
preparation. The self-evaluation is split into two parts. REMINDER, PAGE 6 Concept of Functions

1 In each of the following situations, determine the dependent variable and the independent variable .

Part 1: Explicit Evaluation of Knowledge a) The number of words in a text and the time it takes to write them .
b) The surface area to be painted and the number of paint cans to be bought .
This section contains a series of unrelated questions. Each question targets one or more specific concepts.
c) The average speed of a vehicle and the time it takes to make the journey .

2 Look at the graphs below . Which one(s) do(es) not represent a function?
Part 2: Evaluation of Competencies Explain why .

You will be presented with situational problems similar to those you solved in each of the chapters. You will A) B) C)
y y y
be required to complete tasks involving various concepts in a new context.
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
Instructions 0 x 0 0
2 5 24 23 22 21 1 2 3 4 5 2 5 24 23 2221 1 2 3 4 5 x 2 5 24 23 22 21 1 2 3 4 5 x
22 22 22
23 23 23
• Carefully read each question before answering. 24 24 24
25 25 25

• Note that the use of graphing calculators is permitted, as well as

a quick reference page.
3 In Martha’s kitchen, the tap flows at a rate of 120 ml/s . It takes 3 s to fill a 360 ml glass to the brim . Let t be
• Show each step in your work and calculations. X=0 Y=0
the time (in seconds) required to fill the glass and f(t), the amount of water it contains (in millilitres) .

REFRESHER
• Once completed, correct the self-evaluation using the answer key STAT PLOT F1 TBLSET F2 FORMAT F3 CALC F4 TABLE F5

Y= WINDOW ZOOM TRACE GRAPH

a) Determine the function rule f .
associated with each question. QUIT INS

b) Draw a graph for this function .

2nd MODE DEL
A-LOCK LINK LIST

ALPHA X,T,0,n STAT

TEST A ANGLE B DRAW C DISTR

MATH APPS PRGM VARS CLEAR

c) What is the range of 2 under this function?

()
MATRX D SIN–1 E COS–1 F TAN–1 G π H

x –1
Analyzing Your Performance 1
d) Calculate: f __ and f(1 .5) .
SIN COS TAN ALPHA

I EE J { K } L e M

x –2 , –
10 x N u O v
(
P w
)
Q [ R
2

SELF-EVALUATION
7 8 9 ×
e) Taking the context into account, what is the domain of this function?
LOG

Since this is a self-evaluation, you will analyze your own performance using the evaluation4 grid
ex

5
LN
provided
6
S

–
L4 T L5 U L6 V ] W

at the end. If you are having difficulty, don't hesitate to review the relevant text or contact1 your
2 teacher
+ for
© SOFAD / All Rights Reserved.

RCL X Y Z O MEM

f ) What is the range (codomain) of f ?

L1 L2 L3

© SOFAD / All Rights Reserved.

STO
3
OFF CATALOG i : ANS ? ENTRY SOLVE

help. The Reference column tells you which situations to refer to in the guide. 0 .
ON (–) ENTER

g) How long would it take to fill the glass with:

1) 150 ml of water?
2) 300 ml of water?

4 Using the function f(x) 5 4x 2 6, evaluate the following expressions .

a) f(2) b) f(1 .5) c) f(23) d) f(0)

213 ANSWER KEY PAGE 177 115

SELF-EVALUATION REFRESHER
A Self-evaluation can be found in the first Throughout the Situations, you will come
part of the Complements in Volume 2. across headings entitled Reminders. These
This book is the property of Dickson Joseph.

It allows you to evaluate your acquired sections present concepts seen in a previous
knowledge and the mathematical skills course that are necessary to understand the
you have developed throughout the course. new knowledge or to solve the current
In this way, you will be able to identify the situation.
knowledge that you have mastered and
The Refresher section allows you to use
that for which a revision is necessary before
exercises to review the mathematical rules and
moving on to the Summary Scored Activity.
concepts that are the subject of a Reminder.

knowledge summary MATHEMATICAL REFERENCE

CHAPTER 1
Mathematical Symbols
Step Functions Symbol Meaning Symbol Meaning
A step function is a function that is constant on each of its defining intervals and that jumps from one interval
5 … equals …  Intersection of sets
to the next as the independent variable changes.
The critical values are the endpoints of the intervals where the function varies abruptly. < … is approximately equal to … ∞ Infinity
knowledge summary

As a result, the graph of the function is made up exclusively of horizontal segments called steps. A closed circle  … not equal to … 1∞ Positive infinity
chapter 1

() at the end of a step means that the endpoint is included in the graph of the function. An open circle ()
means the opposite. The range of a critical value always corresponds to the y-coordinate of the closed circle.  Plus or minus 2∞ Negative infinity

 … less than … R Set of real numbers

Example:

© SOFAD / All Rights Reserved.

 … greater than … R1 Set of positive real numbers
Graph of a Step Function Table of Values • The critical values of the
y function are 21 and 1.  … less than or equal to … R2 Set of negative real numbers
5 • The closed endpoints
4 x f(x)  … greater than or equal to … x Change in the x coordinate
3
associated with these critical
[23, 21[ 3 values mean that f(21) 5 22 {l}
2 Empty set y Change in the y coordinate
1
[21, 1[ 2 2 and f(1) 5 2.
2 5 24 23 22 21 1 2 3 4 5 x  Empty set m Slope
22
[1, 4] 2
23
[a] Integer part where a  R mAB Slope of the line AB
knowledge summary

24

25
[a, b] Interval of a to b inclusive || Parallel
chapter 1

[a, b[ Interval including a, but excluding b ^ Perpendicular

Greatest Integer Function f(x) 5 [x]
]a, b] Interval excluding a, but including b |a| Absolute value of a
Greatest integer functions are specific cases of step functions. The integer part of a number, written x,
is the greatest integer less than or equal to that number. ]a, b[ Interval of a to b exclusive dom f Domain of the function f

In the graph of a greatest integer function, all the steps are of the same length, and the jumps between  … belongs to … img f Range of the function f
consecutive steps are equal in height. Step functions are also called staircase functions because of their
MATHEMATICAL

… does not belong to … codom f Codomain of the function f

REFERENCE

obvious resemblance to a staricase. 

© SOFAD / All Rights Reserved.

© SOFAD / All Rights Reserved.

The rule for this greatest integer function is written as follows: f(x) 5 x.  Union of sets

Example:
__ __
f(2.3) 5 2.3 5 2 ; f(√ 2 ) 5 √ 2  5 1 ; f(213.5) 5 213.5 5 214.

121 131

KNOWLEDGE SUMMARY MATHEMATICAL REFERENCE

The full version of the Knowledge Summary is In this section, we present mathematical
located in this section. A printable version is symbols used in the guide and some
also available online. abbreviations of units of measurement.
Reminders of mathematical formulas are
also provided.

VIII HOW THE GUIDE IS STRUCTURED

 TABLE OF
CONTENTS

GLOSSARY ANSWER KEY

CHAPTER 1
Absorbing Element Coefficient of Proportionality
An absorbing element is a number that, by means of In a situation of direct proportionality, this is the SITUATION 1.1
a certain operation, transforms all other numbers or value by which the independent variable must be PAY-AS-YOU-GO CALLS
expressions into itself. multiplied in order to obtain the dependent EXPLORATION 1.1 PAGES 5 TO 6
variable. (Also see Proportional.) 1 In the first contract, the cost is $0.40 for every minute whether 5 a) Independent variable: length of a call (s).
Example: that minute is complete or not. In the second contract, the
b) Dependent variable: cost of a call ($).
The absorbing element of multiplication is 0, since any Example: cost is $0.20 for every complete 30-second block.

ANSWER KEY
CHAPTER 1
number multiplied by 0 will always equal 0. The perimeter P of a square is directly proportional to 6 Cost of a call as a
2 Contract 1: function of its length
the measurement s of its side. The coefficient of Cost
To calculate the cost of a call based on its length (s), ($)
proportionality of this relationship is the value of the
Algorithm P proceed as follows: 2.00
ratio __ , which is equal to 4. Therefore P 5 4s. 1.80
s Divide the number of seconds by 60 to express the
A series of steps that, when followed, will produce 1.60
length (min). 1.40
the desired result regardless of the initial data. 1.20
For 100 s: 100 4 60  1.67.
Completing the Square 1.00
0.80
Then round the number of minutes up to the nearest integer.
Axis of Symmetry A calculation method used to complete a binomial
0.60
1.67 is rounded to 2. 0.40
A straight line that splits a geometric shape into two in order to produce a perfect square trinomial. 0.40 3 2 5 0.80. The cost of a 100-s call is $0.80.
0.20

isometric parts by reflection over the axis. 0 60 120 180 240 300
Contract 2: Length of call
Example: (s)
To calculate the cost of a call based on its length (s),
Binomial By adding 9 to the binomial x2 2 6x, we get the perfect proceed as follows:
square trinomial x2 2 6x 1 9, the factored form of Divide the number of seconds by 30 to find out how many ACQUISITION 1.1A PAGES 7 TO 15
A polynomial with two terms. which is (x 2 3)2. 30-s blocks the call lasted for. 1 a) ]28, 24]; ]24, 4]; ]4, 5] and ]5, 7]
For 100 s: 100 4 30  3.33.
Codomain b) Domain: ]28, 7] Range: {23, 0, 2, 5}
Conjecture Then round the number of 30-s blocks down to the
The set of values that the dependent variable of a nearest integer, which represents the number of complete 2 a) 1) f(10) 5 1 2) f(20) 5 1 3) f(20.5) 5 2 4) f(60) 5 3
A statement that is accepted as true but has not yet blocks only.
function may have. (Synonym: Range of a function.)
been proven. 3.33 is rounded to 3. EXPLANATION: For example, to understand that
Example: 0.20 3 3 5 0.60. The cost of a 100-s call is $0.60.
f(10) 5 1, you have to look at the graph to see the value of
the y-coordinate when the value of the x-coordinate is
Constant (function)
Temperature of a liquid 10 min. In this case, if x 5 10 min, then y 5 $1.00.
as a function of time 3 a) Contract 1:
Temperature A function whose dependent variable can be only
(°C) For 200 s: 200 4 60  3.33. b) The closed circles indicate that the points belong to the
one value. The function rule for this function is

© SOFAD / All Rights Reserved.

6
3.33 min is rounded to 4 min. function, while the open circles indicate that the points

© SOFAD / All Rights Reserved.

5 f(x) 5 a, where parameter a is a real number. On a are excluded.
4 0.40 3 4 5 1.60. The cost of a 200-s call is $1.60.
graph, a constant function is represented by a
3 Contract 2:
horizontal straight line. 3 a) All values greater than 0 g, up to a maximum of 500 g
2
For 200 s: 200 4 30  6.67.
Codomain 1 b) The possible costs are $6, $8 or $12.
0 6.67 is rounded to 6 blocks of 30 s each.
1 1 2 3 4 5 6 7 8 9 10
c) The critical values are 200 and 300.
0.20 3 6 5 1.20. The cost of a 200-s call is $1.20.
2

2 Time
d) 1) 299 g 5 $8.00
2
(min)
3 b) Contract 1:
2) 300 g 5 $8.00
2

The codomain of this function is [22, 4] °C. For 300 s: 300 4 60 5 5. 3) 301 g 5 $12.00
0.40 3 55 2.00. The cost of a 300-s call is $2.00.
e) It weighs between 0 g (exclusive) and 200 g (inclusive).
Contract 2:
For 300 s: 300 4 30 5 10. 4 a) The critical values are {22, 0, 2, 4, 6}.
0.20 3 10 5 2.00. The cost of a 300-s call is $2.00. The range is {26, 23, 0, 3, 6, 9}.

b) The critical values are {24, 0, 4, 8, 12}.

4 30 s The range is {21, 20.5, 0, 0.5, 1, 1.5}.

145
134 GLOSSARY

GLOSSARY ANSWER KEY

Words and expressions written in blue in Toward the end of the guide, you will find
the current text are defined in the Glossary. the Answer Key. It is designed not only
for checking your answers, but also to
complement your learning process.
It contains the answers to questions and
detailed explanations of the approach to
be taken or the reasoning to be used.

EVALUATION GRID Quick RefeRence

Competency 1: Uses strategies to solve situational problems Name of learner:

Evaluation Excellent Very good Good Poor Very poor

criteria A B C D E

1.1 Identifies Identifies nearly Identifies Identifies Identifies very

Indication of all relevant all relevant some relevant little relevant little relevant
an appropriate information. information. information. information. information.
understanding of the
situational problem

1.2 Uses all relevant Uses nearly Uses some Uses few Uses no
Application of strategies. all relevant relevant relevant relevant
© SOFAD / All Rights Reserved.

strategies and strategies. strategies. strategies or strategies or

knowledge* does so with does so with
appropriate to the difficulty. great difficulty.
situational problem.

* The evaluation pertains to the strategies applied.

Competency 2: Uses mathematical reasoning

Evaluation Excellent Very good Good Poor Very poor

criteria A B C D E

2.1 Uses all Uses nearly Uses some Uses necessary Uses necessary
Correct use of necessary all necessary necessary mathematical mathematical
appropriate mathematical mathematical mathematical knowledge knowledge
mathematical knowledge and knowledge and knowledge and with difficulty with great
concepts and obtains all the obtains nearly obttains some and obtains few difficulty and
processes correct results all the correct of the correct of the correct obtains very
results. results. results. few of the
Quick RefeRence

correct results.
EVALUATION

© SOFAD / All Rights Reserved.

2.2 Presents an Presents an Presents an Presents an Presents an

© SOFAD / All Rights Reserved.

GRID

Proper approach that is approach that approach approach that approach that
implementation consistent with is consistent that is fairly is lacking in is very lacking
of mathematical all the selected with nearly all consistent with consistency. in consistency.
reasoning suited to strategies and the selected the selected
the situation. knowledge. strategies and strategies and
knowledge. knowledge.
2.3 Presents an Presents an Presents an Presents an Presents an
Proper organization approach that approach approach that is approach that approach
of the steps in is complete and that is fairly fairly complete is incomplete that is very
an appropriate well organized complete and but not well and not well incomplete and
procedure. and adheres to well organized organized and organized and disorganized
all mathematical and adheres adheres to some adheres to few and adheres
conventions. to nearly all mathematical mathematical to very few
mathematical conventions. conventions. mathematical
conventions. conventions.

* The quick reference must have a maximum length of one page (front) 8 ½ × 11, be handwritten or typed by the learner (minimum font size
12 points, single spaced) and approved by the teacher. Examples provided by the learner and mathematical formulas are permitted. 357
181

EVALUATION GRID QUICK REFERENCE

A competency Evaluation Grid is available You can create your own quick reference guide.
at the end of the guide. After solving an LES, A detachable sheet is provided for this purpose
you are asked to evaluate yourself using this at the end of the guide in Volume 2. You may use
grid. You can then complete the abbreviated this quick reference during the final test.
version at the bottom of each LES.

IX
 TABLE OF
CONTENTS

HEADINGS AND PICTOGRAMS

PIECEWISE FUNCTIONS*

SP 1.1

Refers, if applicable, to optional Invites the student to watch a video

knowledge. It is recognizable by clip on the situational problem.
its paler background.
TASK

Determine the rule of the function Presents the task to be performed

that rounds a number… as part of your Situational Problem.
This book is the property of Dickson Joseph.

REFRESHER EXERCISES
REMINDER PAGE 138, QUESTIONS 1 TO 3 REMEMBER

Representation of… Step Functions

A function is a relationship… A step function is a function that…

Example: Example:
The interval of the numbers 2… The interval of the numbers 2…

© SOFAD / All Rights Reserved.

Presents the mathematical knowledge you
Refers to knowledge you have acquired
will be required to master. This is the
in previous courses and refresher
knowledge prescribed by the study
exercises related to this Reminder.
program.

STRATEGY Adopting…
Presents problem-solving strategies that
When you want to analyze a situation
can be applied in a variety of situations.
involving functions, it can be very helpful to…

X HOW THE GUIDE IS STRUCTURED

 TABLE OF
CONTENTS

DID YOU KNOW?

Closed and open circles were invented due Allows you to discover historical and
to the fact that… cultural information related to the
mathematical concepts being studied.

Tip Provides a tip that simplifies task, or offers

By studying the critical values of a a different way of dealing with the problem
greatest integer function, it is easier of applying the concept being studied.
to get an overview…

CAUTION!
Warns of traps to avoid or exceptions that
Make sure that the intervals that define the may apply to the concept being studied.
parts of the function in your rule are…

ICT Prompts you to complete an online activity

ICT Activity 1.1.2 allows you to observe (GeoGebra or graphing calculator), that will
the effect of making changes to encourage you to explore the concept
parameters a and b on the graph studied using technological tools.
© SOFAD / All Rights Reserved.

of a function. This activity is…

Indicates that you are ready to complete the

ACTIVITY
SCORED

You must now complete Scored Scored Activity designed to assess your
Activity 1. It can be found on the comprehension as you learn. The Summary
course website… Scored Activity is completed at the very end of
the course. These activities are presented in
separate booklets of the guide. Once completed,
you will have to submit your work to your
teacher or tutor who will provide you with
feedback following correction.

XI
 TABLE OF
CONTENTS

CHAPTER 1

Step Functions and Greatest

Integer Functions
Interpreting Numerical
Information

E
ach day, we receive a phenomenal amount of information
in numerical form. Whether on our phones, computers,
tablets or televisions, we are constantly bombarded with
numbers. Often, these numbers are rounded off to make them
This book is the property of Dickson Joseph.

easier to express. Just think about cellphone calls billed to the

next minute or the temperature outside rounded off to the
nearest unit. The same is true for race times, which are rounded
down to the nearest tenth of a second. These rounded values
can be modelled using functions called step functions. In this
chapter, you will learn about this type of function, including how
to describe them and how to represent them using both graphs
and algebra.

© SOFAD / All Rights Reserved.

2 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

CHAPTER 1
SITUATION 1.1
STEP FUNCTIONS

SITUATION 1.1
EXPLORATION
GREATEST INTEGER FUNCTION f (x ) 5 [x ]
GREATEST INTEGER FUNCTION f (x ) 5 a[bx ]
SP 1.1 – P
 ay-as-You-Go Calls p. 4

SITUATION 1.2
GREATEST INTEGER FUNCTION f (x ) 5 a [b (x 2 h)] 1 k
NEAREST INTEGER FUNCTIONS
 easuring the Temperature p. 26
SP 1.2 – M

KNOWLEDGE SUMMARY p. 50
INTEGRATION p. 56
LES
Timekeeping p. 62
© SOFAD / All Rights Reserved.

3
 TABLE OF
ANSWER KEY
CONTENTS

STEP FUNCTIONS
GREATEST INTEGER FUNCTION f(x) 5 [x]
SITUATION 1.1 GREATEST INTEGER FUNCTION f(x) 5 a[bx]

Pay-as-You-Go Calls SP 1.1

Annie, who is studying at

university far from home, has a
cellphone that she rarely uses.
That is why she prefers to pay for
each call she makes based on its
length instead of getting a fixed-
price monthly plan.
This book is the property of Dickson Joseph.

Annie knows that the number of minutes for

each call on her bill has been rounded off. November bill
For example, on November 30, she tried to Type of Length
Date Number Cost ($)
get in touch with a friend who was not there. call (min)
Although this call lasted barely 15 seconds, November 3 819-256-XXX3 Out 4 1.60
1
or __​ ​ ​ of a minute, she was billed for a full November 5 450-231-XXX1 Out 2 0.80
4
minute. Her contract with the phone November 10 450-231-XXX1 In 1 0.40

company is clear in this regard: “Each call November 10 450-231-XXX1 Out 5 2.00

is charged at a rate of $0.40 per minute; November 15 514-634-XXX8 Out 1 0.40

any fraction thereof is rounded up to the November 22 819-321-XXX0 In 3 1.20

nearest minute.” November 23 450-652-XXX8 Out 16 6.40

November 30 514-354-XXX2 Out 1 0.40
Since Annie does not want to pay for time November 30 514-354-XXX2 Out 8 3.20
she did not use, she decides to switch Total 41 16.40
cellphone companies. A new company on

© SOFAD / All Rights Reserved.

the market is offering the following rate in
its contract: “Each call is charged at a rate of
$0.20 per complete block of 30 seconds.” Therefore, if Annie makes a call of less than 30 seconds, there will
be no charge—which seems like a very appealing offer.
Annie has a monthly budget of $30.00 for phone calls. She makes about 10 calls per month to catch up with her
friends and family— 2 or 3 calls a week—so she wants to know what the average length of each call should be.

Using the rule describing the situation, calculate how long Annie’s calls should be on average
TASK

so she does not go over her monthly budget with the new phone company. Support your
calculations with a table of values or a graph.

4 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

EXPLORATION
Answer the questions in this exploration activity to review the mathematical terms related to functions and
identify the characteristics of the function you can use to model Situational Problem 1.1. You will find them
useful when building your graph and determining the function rule.

1 Explain the differences between the two billing rates.

SITUATION 1.1
EXPLORATION
2 To make sure you clearly understand the situation, calculate the cost of a 100-s call on each rate.
Explain your calculations step by step.

3 Apply the procedure you described in your previous answer to calculate the cost of the following calls.
a) A 200-s call

b) A 300-s call
© SOFAD / All Rights Reserved.

STRATEGY Give yourself numerical examples

When you want to analyze a function situation, it is helpful to give yourself numerical examples. For instance, to
determine the rule for a function, you can often simply replace the numbers in the examples with variables.

It is clear that the new company offers a better deal on calls. You probably also noticed that the cost remains
constant for a certain interval of time and then increases by $0.20.

4 How long will a call last before the cost increases by $0.20?

ANSWER KEY PAGE 145 5

 TABLE OF
ANSWER KEY
CONTENTS

5 Define the two variables being correlated in this situation.

a) Independent variable:
b) Dependent variable:

6 Fill in the graph to the right by plotting the cost Cost of a call as a
function of its length
(in dollars) of a call with the new phone company Cost
($)
for each of the following times: 0 s, 30 s, 60 s, 90 s, 2.00
120 s, 150 s, 180 s, 210 s, 240 s, 270 s and 300 s. 1.80

1.60

1.40

1.20
1.00

0.80

0.60

0.40
0.20

0 30 60 90 120 150 180 210 240 270 300

Length of call
This book is the property of Dickson Joseph.

(s)

REMINDER REFRESHER EXERCISES

PAGE 115, QUESTIONS 1 TO 3
The Concept of Functions
A function is a relationship in which each possible value of
Dependent
the independent variable corresponds to a single value of variable Distance travelled
the dependent variable. as a function of time
Distance
The domain of the function is the complete set of possible (m)
20
values of the independent variable. The range (also called the 18
16
codomain) of the function is the complete set of possible 14
12
Range

values of the dependent variable. The relationship between 10

8
the two variables is expressed algebraically by a rule that is 6
4 f(x) 5 2x

© SOFAD / All Rights Reserved.

generally written as f(x) 5 “an algebraic expression of x.” In this 2

rule, x represents a possible value of the independent variable, 0 1 2 3 4 5 6 7 8 9 10

Independent Time
and f(x) is the image of this value defined by the function f. variable (s)

Domain

You have determined some of the values that define the function representing the situation. These values
represented calls consisting of full 30-second blocks. In the following acquisition activity, you will build a
graph illustrating the function associated with the situation, which will take all fractions of minutes into
account. It will also help you define the rule you need to complete the task.

6 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Mathematical knowledge
ACQUISITION A targeted:
•• interpreting a step
function
•• determining the rule for
1. Step Functions a greatest integer
function from its graph
In order to round off numbers of any value, it is important
to study a type of function called step functions.
•• determining a missing
value algebraically, using
A step function is a discontinuous function that is constant the rule of a greatest

SITUATION 1.1
integer function

ACQUISITION A
over certain intervals called steps.

1 Consider the following graph of a step function.

y
5
4
3
2
1

28 27 26 25 24 3 22
2 21 1 2 3 4 5 6 7 x
22
23

24

5
2

a) On which intervals of x is each step in this step function?

b) What are the domain and the range of the function illustrated above?
Domain: Range:

REMINDER REFRESHER EXERCISES

PAGES 115 TO 117, QUESTIONS
4 TO 9
© SOFAD / All Rights Reserved.

Representing Intervals
An interval is a set consisting of all the real numbers found between two endpoints. It may be
represented in words, in a graph, in set-builder notation or in interval notation.

Example:
The interval of the numbers from 2 (included) to 5 (excluded) may be represented
• graphically:
2 5 R

• in set-builder notation: {x  R | 2  x , 5}

• in interval notation: [2, 5[

ANSWER KEY PAGE 145 7

 TABLE OF
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2 Now consider the meaning of the open and closed circles at the Cost of parking
endpoints of each step in a step function. as a function of
parking time
Cost
To the right is a graph of a step function representing the ($)
4
cost of parking on a street in Montréal.
3
a) Using the graph, evaluate the following functions. 2

1) f(10) 5 2) f(20) 5 1

0 20 40 60 80 100
3) f(20.5) 5 4) f(60) 5 Time
(min)

b) What do you think the open and closed circles mean?

REMEMBER

Step Functions
A step function is a function that is constant on each of its defining intervals and that jumps from one
interval to the next as the independent variable changes.
This book is the property of Dickson Joseph.

The critical values are the endpoints of the intervals where the function varies abruptly.
As a result, the graph of the function is made up exclusively of horizontal segments called steps.
A closed circle () at the end of a step means that the endpoint is included in the graph of the function.
An open circle () means that the endpoint is not included. The image of a critical value always
corresponds to the y-coordinate of the closed circle.

Example:
Graph of a step function Table of values • The critical values of the
y function are 3 and 5.
The function is constant
on each step
8 • The closed endpoints
7
associated with these
6
Jumps
x f(x)
5 critical values mean that
[0, 3] 4
4 f(3) 5 4 and f(5) 5 2.

© SOFAD / All Rights Reserved.

3 ]3, 5] 2
2 Critical
values ]5, 8] 7
1

0 1 2 3 4 5 6 7 8 x

8 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
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PRACTISE
Postage rate
3 The graph to the right illustrates postage rates for a as a function
of parcel weight
small parcel to the United States, depending on its Rate
($)
weight (up to 500 g). 12
10
a) What values can you associate with the weight
8
of a parcel in this context?
6
4

SITUATION 1.1
ACQUISITION A
b) What are the possible costs of mailing a parcel? 2

0 100 200 300 400 500

Weight
c) What are the critical values of the function? (g)

d) How much would it cost to mail a parcel of each of the following weights?
1) 299 g 2) 300 g 3) 301 g
e) How much does a parcel weigh if it costs $6 to send?

4 Determine the critical values and images of each of the functions represented in the following graphs.
a) b)
y
10

8 y

6
2
4
1
2

2 8 6
2 4
2 2
2 2 4 6 8 10 x 2 8 2 4 4 8 12 16 x
22 1
2

24
2
2

26
28
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ANSWER KEY PAGE 145 9

 TABLE OF
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2. Greatest Integer Functions

You may have noticed that the steps in the two previous graphs were similar. In fact, in both cases, the steps
were all of the same length and the value of the jump between steps was also the same. This type of step
function is called a greatest integer function.

2.1 Greatest Integer Function f(x) 5 [x]

To interpret these functions correctly, you will discover a new use for the symbols [x].

5 Below are a few calculations in [x] notation.

[5] 5 5 [14.31] 5 14
__
​​[​√ 7
 ​ ]​​ 5 2 [27.1] 5 28

Since this notation is known as the integer part of x, what conjecture can you make about its use?

REMEMBER
This book is the property of Dickson Joseph.

Greatest Integer Function f(x) 5 [x]

Greatest integer functions are specific cases of step functions. The integer part of a number x (written x)
is the greatest integer less than or equal to that number.
In the graph of a greatest integer
f(x) Steps all of
function, all the steps are of the Example: equal length
same length, and the jumps 4
f(1.1) 5 1.1 5 1
between consecutive steps are 3

equal in height. Step functions are f(p) 5 p 5 3 2

also called staircase functions f(22.6) 5 22.6 5 23 1
because of their obvious 2 2.6

resemblance to a staircase. f(212.1) 5 212.1 5 213 4

2 3
2 2
2 1
2 1 2 3 4 x
2 1 1.1 π
The rule for this greatest integer 2 2
function is written as follows:

© SOFAD / All Rights Reserved.

2 3
f(x) 5 x. 2 4
Jumps all of
equal height

PRACTISE

6 Calculate the values of the following expressions.

ICT
a) [12] 5
ICT Activity 1.1.1 shows you how to use a
b) [5.99999] 5 graphing calculator to find the integer part of a
[]
c) ​​ __
1
​   ​ ​​ 5
2
number. See this activity on portailsofad.com.

d) [23.4 1 1.2] 5

10 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
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CONTENTS

2.2 Greatest Integer Function f(x) 5 a[bx]

To expand on greatest integer functions, certain parameters may be added to the basic formula. The first two
parameters you will study are a and b, with a rule written as follows: f(x) = abx. The following questions will
help you understand what each parameter means.

7 Consider the three graphs below. Then fill in the missing information.

f(x)f(x)
 2[x]2[x]
f(x)
  2[x] g(x)g(x)
 [2x][2x]
g(x)
  [2x] h(x)h(x)
 3[0.5x]
3[0.5x]
h(x)
  3[0.5x]
y y y y y y y y y

SITUATION 1.1
6 6 6 6 6 6 6 6 6

ACQUISITION A
4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 2 2

 6 6 4 642 42 2 2 2 4 24 6 4x6 x 6 x

  6
 6 4 642 42 2 2
  2 4 24 6 4x6 x 6 x 6
   6 4 642 42 2 2 2 4 24 6 4x6 x 6 x
2 2 2 2 2 2 2 2 2

 4  4  4  4  4  4 4
 4
 4

 6  6  6  6  6  6 6
 6
 6

a5 2 b5 1 a5 b5 a5 b5
Length of one step: 1 Length of one step: Length of one step:
Height of one jump: 2 Height of one jump: Height of one jump:

STRATEGY Recognize parameters quickly

Regardless of the type of function, relating the parameters of the function rule and the characteristics of its graph
makes it easier to go from one representation to the other. You can then quickly draw the graph from the rule or,
conversely, determine the rule from the graph.

8 Now try to determine the rule for the function

y
represented in the graph to the right.
© SOFAD / All Rights Reserved.

a) What is the height of each jump between two steps? 4

b) What is the length of each step?

28 2 6 4
2 2 2 2 4 6 8 x
c) Determine the rule for the function. 2 2

2 4

2 6

ANSWER KEY PAGE 146 11

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REMEMBER

Greatest Integer Function f(x) 5 a[bx]

In a greatest integer function of the form f(x) 5 a[bx] :
• The first closed endpoint is at (0, 0).
• Parameter a determines the height of a jump; this height is equal to |a| .
1
• Parameter b determines the length of a step; this length is equal to __
​​  ​​ .
|b|

NOTE: The symbol |x| represents the absolute value of a number x, that is, the value of the number without regard
to its sign. Therefore, |2| 5 2 and |22| 5 2.

To determine the rule for a greatest integer function from its graph or a table of values,
you must do the following:
• Determine the value of parameter a by calculating the height of a jump between two
consecutive steps.

• Determine the value of parameter b by calculating the length of a step; parameter b will
1
be equal to _________________
This book is the property of Dickson Joseph.

​​   ​​.
Length of step

Example:
Graph of a greatest integer function Table of values
y

6
844
4
x f(x)
Step
[28, 24[ 24
2 12
[24, 0[ 22
12
8
2 26 24 22 2 4 6 8 x [0, 4[ 0
2
2
12
202 [4, 8[ 2
Jump
4
2

© SOFAD / All Rights Reserved.

82454
6
2

Height of one jump: 2 Length of one step: 4

1
Value of parameter a: 2 Value of parameter b: __
​​   ​​
4
Rule: f(x) 5 2​​ __[ ]
1
​   ​  x ​​
4

12 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
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CONTENTS

PRACTISE

9 [ ]
Let the functions f(x) 5 0.5​​ __
1
​  ​  x ​and g(x) 5 22x. Determine the values of parameters a and b
4
for each function. Then determine the length of the steps and height of the jumps in the graph
of each function.

f(x) g(x)
a5 b5 a5 b5
Length of one step: Length of one step:

SITUATION 1.1
ACQUISITION A
Height of one jump: Height of one jump:

10 Let the function h(x) 5 30.5x.

a) What are the values of parameters a and b?
b) What are the critical values of the function?
c) Complete the table of values below by filling in the x column with the intervals whose
endpoints are critical values. Then plot the corresponding graph.

x h(x) 5 30.5x

[0, 2[

[2, 4[
© SOFAD / All Rights Reserved.

Production of lavender oil

11 The graph to the right illustrates the production as a function of the quantity
of flowers used
of a company that makes artisanal essential oils Quantity of oil
produced
4
from lavender. It takes 0.5 kg of lavender flowers (ml)

to produce 1 ml of essential oil. 3

2
Model the situation by finding the rule that
1
describes the lavender oil production.
0 0.5 1 1.5 2 2.5
Quantity of
lavender flowers
(kg)

ANSWER KEY PAGE 146 13

 TABLE OF
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3. From the Rule to the Values of a Greatest Integer Function

So far, the analysis in this section has focused mainly on the graphical representation of the greatest integer
function. However, using the rule to find missing values is often more efficient.

12 Let the function f(x) 5 50.25x. Find each of the following values.
a) f(4)

b) f(26)

c) x, if f(x) 5 20
This book is the property of Dickson Joseph.

d) x, if f(x) 5 240

e) x, if f(x) 5 6

© SOFAD / All Rights Reserved.

Tip
Isolating x in the equation 2 5 x is the same as asking which values of x will have an integer part of 2.
Since the integer part of a number is the greatest integer less than or equal to that number, you can
conclude that all values between the number in question and the next integer will be part of the
solution—in this case, 2, 3 or 2  x  3.

14 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
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CONTENTS

REMEMBER

From the Rule to the Values of a Greatest Integer Function

• To determine a value of f(x) when x is known, simply replace x in the rule with its value and follow the
order of operations. In this case, the integer part symbol has the same priority as brackets in the
order of operations.
• To determine a value of x when f(x) is known,
you must proceed step by step: CAUTION!

SITUATION 1.1
1) Isolate the integer part.

ACQUISITION A
If the integer part is equal to a value
2) Determine the interval that satisfies the integer part. that is not an integer, there is no
This interval always takes the form z, z 1 1, where z solution. In other words, there is no
is an integer. step at this height.

3) Isolate x in the expression contained in the integer part.

Example:
Let the function f(x) 5 30.2x.
If the value of x is 12, If the value of f(x) is 212, If the value of f(x) is 5, then:
then: then:
5 5 30.2x
f(12) 5 30.2 × 12 212 5 30.2x _
1.​​6​​ 5 0.2x
f(12) 5 32.4 24 5 0.2x
Since_ no number has an integer part
f(12) 5 3 3 2 5 6 24  0.2x , 23 of 1.​​6​​, there is no solution.
220  x , 215

PRACTISE

13 Let the function f(x) 5 4 [0.5x].

© SOFAD / All Rights Reserved.

a) Evaluate the following expressions.

1) f(4) 5 2) f(27) 5
b) Determine the value or values of x for each equation.
1) f(x) 5 4 2) f(x) 5 6

In this acquisition activity, you discovered a new function called the greatest integer function. You now know
how to determine a rule of the form f(x) 5 a[bx] and represent this function using a graph or table of values.
You can now solve Situational Problem 1.1: “Pay-as-You-Go Calls.”

ANSWER KEY PAGE 146 15

 TABLE OF
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SOLUTION
You can now solve Situational Problem 1.1. STEP FUNCTIONS
GREATEST INTEGER FUNCTION f(x) 5 [x]
SITUATION 1.1 GREATEST INTEGER FUNCTION f(x) 5 a[bx]

Pay-as-You-Go Calls SP 1.1

Annie, who is studying at

university far from home, has a
cellphone that she rarely uses.
That is why she prefers to pay for
each call she makes based on its

Using the rule describing the situation, calculate length instead of getting a fixed-
price monthly plan.

how long Annie’s calls should be on average so Annie knows that the number of minutes for
TASK

each call on her bill has been rounded off. November bill
For example, on November 30, she tried to

she does not go over her monthly budget with the

Type of Length
Date Number Cost ($)
get in touch with a friend who was not there. call (min)
Although this call lasted barely 15 seconds, November 3 819-256-XXX3 Out 4 1.60
1
or __ of a minute, she was billed for a full November 5 450-231-XXX1 Out 2 0.80

new phone company. Support your calculations

4
minute. Her contract with the phone November 10 450-231-XXX1 In 1 0.40

company is clear in this regard: “Each call November 10 450-231-XXX1 Out 5 2.00

is charged at a rate of $0.40 per minute; November 15 514-634-XXX8 Out 1 0.40

with a table of values or a graph. any fraction thereof is rounded up to the

nearest minute.”
November 22
November 23
November 30
819-321-XXX0
450-652-XXX8
514-354-XXX2
In
Out
Out
3
16
1
1.20
6.40
0.40
Since Annie does not want to pay for time November 30 514-354-XXX2 Out 8 3.20
she did not use, she decides to switch Total 41 16.40
cellphone companies. A new company on

© SOFAD / All Rights Reserved.

the market is offering the following rate in
its contract: “Each call is charged at a rate of
$0.20 per complete block of 30 seconds.” Therefore, if Annie makes a call of less than 30 seconds, there will
be no charge—which seems like a very appealing offer.
Annie has a monthly budget of $30.00 for phone calls. She makes about 10 calls per month to catch up with her
friends and family— 2 or 3 calls a week—so she wants to know what the average length of each call should be.

Using the rule describing the situation, calculate how long Annie’s calls should be on average

TASK
so she does not go over her monthly budget with the new phone company. Support your
This book is the property of Dickson Joseph.

calculations with a table of values or a graph.

4 CHAPTER 1 – Step Functions and Greatest Integer Functions

SITUATIONAL PROBLEM FROM PAGE 4

Summary of the Facts

• Annie has a monthly budget of $30.00.

• On average, she makes 10 calls a month.
• A new company is offering a contract that reads as follows:
“Each call is charged at a rate of $0.20 per complete block of 30 seconds.”

Representation of the Situation

(Table of values or graph)

© SOFAD / All Rights Reserved.

16 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Solution

SITUATION 1.1
SOLUTION
© SOFAD / All Rights Reserved.

Function rule:
Answer:

ANSWER KEY PAGE 147 17

 TABLE OF
ANSWER KEY
CONTENTS

Mathematical knowledge
ACQUISITION B targeted:
•• analyzing the effect of
reversing the signs of
parameters a and b
1. The Effect of Reversing the
Signs of Parameters a and b
In the task about cellphone contracts, billing practices were one of the main differences between the two
companies: one company billed by rounding up to the next integer, while the other charged by the integer
but did not count incomplete blocks of time. This type of difference changes the appearance of the graph as
well as the rule of the greatest integer function.
To visualize the effect of reversing the sign of parameter a or b, a graph of the function f(x) 5 30.5x is plotted
on a Cartesian plane below. The accompanying questions will help you understand how the graph reflects
the changes in the parameters.

1.1 Reversing the Sign of Parameter a

1 If parameter a becomes negative in the initial function f(x) 5 30.5x, the function rule becomes
f1(x) 5 230.5x.
This book is the property of Dickson Joseph.

a) Evaluate the following expressions.

1) f1(23) 5
2) f1(22) 5 Tip
3) f1(21) 5 By studying the critical values of a greatest integer
4) f1(0) 5 function, it is easier to get an overall picture of the
function and, consequently, to extract information on
5) f1(1) 5 the impact of the parameters.
6) f1(2) 5

b) Plot the function f1(x) on the same

y
Cartesian plane as function f(x).
8
f(x)

© SOFAD / All Rights Reserved.

c) What type of reflection did the graph
of function f1(x) undergo when the sign 6
of parameter a was reversed?
4

2
d) Is this an increasing function or a
decreasing function?
28 26 2 4 2
2 2 4 6 8 x

2
2

4
2

6
2

8
2

18 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

1.2 Reversing the Sign of Parameter b

2 If parameter b becomes negative in the initial function f(x) 5 30.5x, the function rule becomes
f2(x) 5 320.5x.
a) Evaluate the function for a few values of x.

b) Plot the function f2(x) on the Cartesian plane. y

SITUATION 1.1
ACQUISITION B
8
c) What type of reflection did the function f(x) f(x)

undergo when the sign of parameter b was 6

reversed?
4

d) Is the function increasing or decreasing? 2

2 8 2 6 2 4 2 2 2 4 6 8 x
2 2

2 4

2 6

2 8

1.3 Reversing the Signs of Parameters a and b

3 If both parameters become negative, the function rule becomes f3(x) 5 2320.5x.
a) Plot the function f3(x) on the Cartesian plane.
b) Check the accuracy of your graph by
y
calculating the following values.
8
f(x)
© SOFAD / All Rights Reserved.

1) f3(1) 5
6
2) f3(21) 5
3) f3(2) 5 4

4) f3(22) 5 2

ICT
28 2 6 2 4 2
2 2 4 6 8 x

2
2

In ICT Activity 1.1.2, you can observe

4
2

the effect of changing parameters

a and b on the graph of a function. 6
2

See this activity on portailsofad.com.

8
2

ANSWER KEY PAGE 147 19

 TABLE OF
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REMEMBER

The Effect of Reversing the Sign of Parameter a or b

• Reversing the sign of parameter a produces a reflection over the x-axis.
• Reversing the sign of parameter b produces a reflection over the y-axis.

y y

1 1

1 x 1 x

Effect of reversing the Effect of reversing the

sign of parameter a sign of parameter b

Direction of Change
You can now draw the following conclusions about the effect on function type:
This book is the property of Dickson Joseph.

• If parameters a and b have the same sign, the function is increasing.

• If parameters a and b have opposite signs, the function is decreasing.

Direction of the Step

If parameter b
If b  0, the steps go from left to right. is positive

If parameter b
If b  0, the steps go from right to left. is negative

PRACTISE

© SOFAD / All Rights Reserved.

4 Graph the following functions.
a) f(x) 5 2.522x b) f(x) 5 2320.25x

20 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

5 Below are graphs of two greatest integer functions. Determine the rule for each function.

y y y y

6 6 6 6

5 5 5 5

4 4 4 4

3 3 3 3

2 2 2 2

SITUATION 1.1
ACQUISITION B
1 1 1 1

2 3 2.5232222.5
2 21.5222121.5
20.521 20.5
0.5
21
1 0.51.5 1 2 1.52.5 2 3 2.5 x 3 x 2 3 22.5232222.5 222121.5
21.5 20.521 2 0.50.5
21
1 0.51.5 1 2 1.52.5 2 3 2.5 x 3 x
21 21

22 22 22 2 2
23 23 23 2 3
24 24 24 2 4
25 25 25 2 5
26 26 26 2 6

f(x) 5 g(x) 5

The first learning situation for this chapter ends here. The exercises that follow will help you consolidate
your learning.

DID YOU KNOW?

_
The numbers 1 and 0.​9
​ ​​
The use of closed and open endpoints is due to the fact that it is impossible to name the number that
immediately precedes _ another number. For example, it may seem reasonable to believe that the number
right before 1 is 0.​​9​​, but these numbers are actually equivalent! Consider the following demonstration.
_
x 5 0.​​9​
_​ Use as a starting point.
10x 5 9.​​9​
_​ Multiply both sides of the equation by 10.
10x 2 x 5 9.​​9​ ​ 2 x Subtract the same quantity from both sides of the equation, namely, x.
© SOFAD / All Rights Reserved.

_ _ _
10x 2 x 5 9.​​9​​2 0.​​9​​ Replace x by its equivalent starting value, namely, 0. 9.
9x 5 9 Perform the subtractions.
x51 Divide both sides of the equation by 9.
_
Since x is_ equal to both 1 and 0.​​9​​at the same time, according to the first and last lines of the demonstration,
1 and 0.​​9​​must be equivalent!

ANSWER KEY PAGE 148 21

 TABLE OF
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CONSOLIDATION

1 Let the function f(x) 5 6[0.25x].

a) Evaluate the following expressions.
1) f(8) 5
2) f(23) 5
3) f(5.8) 5
4) f(p) 5
b) What is the range of f?

c) What are the critical values of f?

d) Graph the function.

This book is the property of Dickson Joseph.

2 Let the function f(x) 5 210[20.1x].

a) Evaluate the following expressions.
1) f(212) 5 2) f(26) 5
b) Is the function increasing or decreasing? Justify your answer.

c) Based on the values of parameters a and b in the function rule,

identify the following.
1) The direction of the steps in the graph

© SOFAD / All Rights Reserved.

2) The length of the steps

3) The height of the jumps

d) Graph the function.

22 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
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CONTENTS

3 The same horizontal segments AB and CD have been drawn on the following three Cartesian planes.

Plane 1 Plane 2 Plane 3

y y y

6 6 6
C D C D C D
4 4 4
A B A B A B
2 2 2

SITUATION 1.1
CONSOLIDATION
2 6 4
2 2
2 2 4 6 x 2 6 4
2 2
2 2 4 6 x 6
2 4
2 22 2 4 6 x
2
2 2
2 2
2

4
2 4
2 4
2

6
2 6
2 6
2

a) Draw the new function produced by the following geometric transformations:

• Plane 1: a reflection on the x-axis
• Plane 2: a reflection on the y-axis
• Plane 3: a reflection on the y-axis followed by a reflection on the x-axis
b) Which change or changes in the signs of parameters a and b in the rule for a greatest integer function
produce the same effect as each of the geometric transformations mentioned in a)?
Plane 1:

Plane 2:

Plane 3:

4 Explain why the following equation is impossible to solve: 6 5 40.3x.

© SOFAD / All Rights Reserved.

5 Determine the value of x in each of the following equations.

a) 2x] 5 26 b) 25 5 250.1x]

ANSWER KEY PAGE 148 23

 TABLE OF
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6 Find the values of parameters a and b, then the rule, for each of the functions represented below.
a) b) x g(x)
y
6 ]215, 210] 40
]210, 25] 20
4
]25, 0] 0
2
]0, 5] 220

2 6 4
2 2
2 2 4 6 x
]5, 10] 240

2
2

4
2

6
2

a5 b5 a5 b5
f(x) 5 g(x) 5

7 A sign outside a hospital indicates the rates for using the parking lot.
a) How much will you pay if you park for the following times?
This book is the property of Dickson Joseph.

1) 1 h 40 min
2) 2 h
b) Graph the relationship between length of stay and cost of parking.
Cost as a function
of length of stay
Cost
($)
12

10

© SOFAD / All Rights Reserved.

2

0 1 2 3 4
Length of stay
(h)

c) Let C(x) be the cost in dollars, where x represents the length of stay in hours.
Determine the function rule for a length of stay between 0 and 4 hours.

d) What is the range of the function?

24 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

8 To run 2000 m, you have to do five laps of a 400-m track.

Each of these laps may be numbered from 1 to 5. It is thus
possible to say that a runner is on his first lap if the distance
he has run so far, in metres, falls within the interval ]0, 400].
It is only when he has passed the 400-m mark that he begins
his second lap.
a) Which lap is the runner currently on when he has run
exactly each of the following distances?

SITUATION 1.1
CONSOLIDATION
1) 600 m
2) 900 m
3) 1200 m
b) A runner is on his fourth lap. What distance may he have run?

c) Let L(x) be the whole number corresponding to the number of the lap the runner is currently on for a
distance covered of x metres. The function L is a step function. Use a table of values and a graph to
represent the function.
© SOFAD / All Rights Reserved.

d) What is the rule for the function L? Justify your answer.

e) Check your answers from part a) by evaluating the following expressions using your rule.
1) L(600) 5 2) L(900) 5 3) L(1200) 5
f ) Describe the domain and the range of the function L, taking the context into account.
dom L 5 ran L 5

ANSWER KEY PAGE 149 25

 TABLE OF
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CONTENTS

GREATEST INTEGER FUNCTION f(x) 5 a[b(x 2 h)] 1 k

SITUATION 1.2 NEAREST INTEGER FUNCTIONS

Measuring the Temperature SP 1.2

A weather website shows the current

temperature in a few towns in the
Laurentians and the surrounding
areas. All of Québec’s weather stations
measure temperature precisely, to the
hundredth of a degree. However, tem-
peratures are usually broadcast to the
public as whole numbers.
This book is the property of Dickson Joseph.

These numbers are automatically rounded off by data processing software, without any human intervention.

© SOFAD / All Rights Reserved.

All temperatures measured across Québec are thus processed in a fraction of a second, using a real function.
It is important to understand how temperatures are rounded off in order to interpret them properly. For
example, as you certainly know, the freezing point of water is 0 °C; if the temperature rises above this point,
snow and ice melt. For the people who run the ski resort at Mont-Tremblant, it is important to know the real
temperature so they can plan which slopes should be open or shut and inform skiers. In this learning
situation, you need to provide Mont-Tremblant’s operators with a mathematical tool for making sound
decisions based on the temperature.

You must determine the function rule for rounding a number to the nearest integer so that the
TASK

Mont-Tremblant ski resort can automate its information process. You must also determine the
interval containing the real temperature at Mont-Tremblant when the weather report says it is
0 °C outside. Show your work and support your answer with graphs.

26 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

EXPLORATION
Answer the questions in this exploration activity to define the type of rounding used for temperatures so that
weather reports are as accurate as possible. The answers will help you complete the task in this learning situation.

1 In the illustration on the previous page, the real temperature at Sainte-Agathe-des-Monts was 2.45 °C,
but it was rounded off to 2 °C. To determine the type of rounding used, answer the following two questions.

SITUATION 1.2
EXPLORATION
a) Round the following values
Type of rounding 2.05 °C 2.50 °C 2.65 °C
according to the type of
1) Up to the nearest integer
rounding required.
2) Down to the nearest integer
3) To the nearest integer

STRATEGY Compare to establish a rule

Using a table to compare numerical values associated with several functions makes it easier to determine the
characteristics of each function and to establish a function rule.

b) Which type of rounding ensures that the difference between the rounded value and the initial
number is always as small as possible? Give examples to justify your answer.

REMINDER REFRESHER EXERCISES

PAGE 118, QUESTION 11
Rounding to the Nearest Integer
In general, rounding to the nearest integer can be described by the following rule:
• If the number in the tenths position is 0, 1, 2, 3 or 4,
it is rounded down (to the integer below).
Example: 2.4 becomes 2.
© SOFAD / All Rights Reserved.

• If the number in the tenths position is 5, 6, 7, 8 or 9,

Example: 2.6 becomes 3.
it is rounded up (to the integer above).

CAUTION!
Rounding to the nearest integer provides more accurate information than simply rounding up or down. When
rounding to the nearest integer, the difference between the number and its rounded value is within the interval
[0, 0.5], so it is never more than 0.5. In the other types of rounding, the difference is within the interval [0, 1[,
meaning that it may be greater than 0.5, as you saw in the table for question 1.

ANSWER KEY PAGE 150 27

 TABLE OF
ANSWER KEY
CONTENTS

2 Apply the rule for rounding to the Temperature rounded

to nearest integer based
nearest integer and graph the on real temperature
Rounded
temperatures between 0 °C and 3 °C. temperature
(°C) 3

0 1 2 3
Real temperature
(°C)

3 As you can see, this graph resembles the graph for the greatest integer function f(x) 5 [x].
However, there is an essential difference: the critical values. What are they?
This book is the property of Dickson Joseph.

To do the task in Situational Problem 1.2, you must complete the graph above by incorporating data with
negative values. The following acquisition activity will help you identify the function rule for rounding real
temperatures to the nearest integer, whether they are positive or negative.

© SOFAD / All Rights Reserved.

28 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Mathematical knowledge
ACQUISITION A targeted:
•• discovering greatest
integer functions of the
form f(x) 5 a [b(x 2 h)]
1. Greatest Integer Function 1k
•• graphing greatest integer
f(x) 5 a[b(x 2 h)] 1 k functions of the form
f(x) 5 a [b(x 2 h)] 1 k
In this section, you will extend your study of greatest integer
functions to those of the form f(x) 5 ab(x – h) 1 k. You will •• determining the rule for

SITUATION 1.2
a transformed greatest

ACQUISITION A
begin by studying the effect of changing each of the four integer function using its
parameters (a, b, h and k). graph
In Situational Problem 1.1, you studied the effect of reversing •• studying nearest integer
functions
the signs of parameters a and b. In this section, you will
review the roles of parameters a and b but focus more on
the effect of changing their numerical values; you will also
see the effects of two more parameters : h and k.
Below are five questions about laboratory tests to determine the dosage of a new drug for young children.
These questions will help you understand the four possible parameters in this greatest integer function.

1 A pharmaceutical company is studying the appropriate doses of a new drug for children aged 0 to 6. Its
initial study suggests a dosage of 1 ml for each full year of age. The following graph shows these doses.

Dose of medication
based on child’s age
Dose

DID YOU KNOW?

(ml)
5

4
An appropriate dose of a particular
3 medication is often determined
2
by a person’s weight. In young
children, the correlation between
1 age and weight is very strong. This
is why both figures are indicated on
0 1 2 3 4 5 the labels of children’s medicine.
© SOFAD / All Rights Reserved.

Age
(y)

a) Which type of function does this graph illustrate?

b) In this context, what does the first step represent?

ANSWER KEY PAGE 150 29

 TABLE OF
ANSWER KEY
CONTENTS

2 After conducting certain clinical studies, the company finds that the initial dose is not sufficient and
suggests doubling it.
a) Graph this new function.
b) What is the function rule for the new dosage?

c) What visual change did the initial graph undergo

to produce this new graph?

3 Once again, the doses are not correct. The company therefore conducts more clinical studies and
suggests cutting the time between dose increases by half.
a) Graph this new function.
This book is the property of Dickson Joseph.

b) What is the function rule for the new dosage?

c) What visual change did the previous graph undergo

to produce this new graph?

© SOFAD / All Rights Reserved.

30 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

4 Further changes are needed. The company finds that the doses are still too low. It also discovers that
the concern about the risk to very young children is unfounded. It therefore suggests increasing all
doses by 1 ml.
a) Graph this new function.
b) What is the function rule for the new dosage?

c) What visual change did the previous graph undergo

SITUATION 1.2
to produce this new graph?

ACQUISITION A
5 Finally, the doses seem adequate, but not appropriate to the children’s ages. The company therefore
suggests changing the age for each dose (for example, a child aged 0.5 year should receive the dose
previously given to a one-year-old; a child aged 1.5 years should receive the dose previously given to a
two-year-old, and so on). Therefore, simply adding 0.5 year to the child’s age will give the correct dose.
a) Graph this new function.
b) What is the function rule for the new dosage?

c) What visual change did the initial graph undergo

to produce this new graph?
© SOFAD / All Rights Reserved.

REMEMBER

The Effect of Changing Parameters a, b, h and k

• Increasing |a| increases the height of the jump. The function is said to undergo a vertical stretch (or
vertical dilation). Decreasing |a| has the opposite effect: the function undergoes a vertical compression.
• Increasing |b| decreases the length of the step. The function is said to undergo a horizontal
compression. Decreasing |b| has the opposite effect: the function undergoes a horizontal stretch.
• Increasing parameter h results in a horizontal translation to the right. Decreasing h has the opposite
effect: a translation to the left.

ANSWER KEY PAGE 151 31

 TABLE OF
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CONTENTS

• Increasing parameter k results in an upward vertical translation. Decreasing k has the opposite effect:
a downward translation.

Example:
In the function g(x) 5 2​​ __
23.
1
[ ]
​  ​  (​x 2 1) ​​ 2 3, the value of parameter h is 1 and the value of parameter k is
2

The graph of f(x) 5 2​​ __

1
[2 ]
​  ​  x ​will undergo a horizontal translation of 1 unit to the right

and a vertical translation of 3 units down. Consequently, there is a closed endpoint at (1, 23).

Parameter a Parameter b
Increasing parameter a results in a vertical stretch Decreasing parameter b results in a horizontal
relative to the greatest integer function f(x) 5 x]. stretch.
f(x)  [x] and g(x)  2[x] f(x)  2[x] and g(x)  2[ 12 x]
y y

5 5
4 4
3 3
2 2
1 1
This book is the property of Dickson Joseph.

5 4 3 2 1 1 2 3 4 5 x 5 4 3 2 1 1 2 3 4 5 6 x
2 2

3 3

4 4

5 5

Parameter h Parameter k
Increasing parameter h results in a horizontal Decreasing parameter k results in a vertical
translation of 1 unit to the right. translation of 3 units down.
f(x)  2[ 12 x ] and g(x)  2[ 12 (x  1)] f(x)  2[ 12 (x  1)] and g(x)  2[ 12 (x  1)]  3
y y
5 5
4 4
3 3
2 2
1 1

© SOFAD / All Rights Reserved.

 4 3 2 1 1 2 3 4 5 6 7 x 2 1
 1 2 3 4 5 6 7 8 9 x
2 2

3 3

4 4

5 5

PRACTISE

6 What changes has the graph of function g(x) 5 −25(x 1 1)] 2 4 undergone relative to the greatest
integer function f(x) 5 x]?

32 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
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CONTENTS

2. Graphing a f(x) 5 a[b(x 2 h)] 1 k Function

To graph a greatest integer function written in its standard form, you must identify the four parameters and
represent their roles on the graph. Follow the steps below.

REMEMBER

Graphing a f(x) 5 a[b(x 2 h)] 1 k Function

To graph a greatest integer function of the form f(x) 5 ab(x 2 h) 1 k, you must determine

SITUATION 1.2
ACQUISITION A
the following information from the parameters:
1) The coordinates of a closed endpoint: (h, k)
1
2) The length of one step: ______
​​   ​​
|​ b |​
3) Whether the closed endpoints are on the left (b . 0) or the right (b , 0)
4) Whether the function is increasing (a and b have the same sign) or decreasing
(a and b have opposite signs)
5) The height of the jumps: |a|

Example:
In the function f(x) 5 20.25(x 1 3) 1 2, you know the following information:
• The coordinates of one of the closed endpoints are (23, 2).
• The steps are 4 units in length because b 5 0.25.
• The steps go from a closed endpoint to an open endpoint because b is positive .
• The function is increasing because a and b have the same sign.
• The jumps are 2 units in height because a 5 2.

With this information, you can now graph the function f(x) 5 20.25(x 1 3) 1 2.
y

10

8
© SOFAD / All Rights Reserved.

6 CAUTION!
4
The way you read parameter h may cause
2 problems. Since the greatest integer function is
written f(x) 5 ab(x – h) 1 k, it is important to be
210 8
2 6
2 2 4 2
2 2 4 6 8 10 x sure of the sign of parameter h. For example, in
22
f(x) 5 20.25(x 1 3) 1 2, parameter h 5 23, not
24
3, because in fact f(x) 5 20.25(x − (23) 1 2.
26
28

2 10

ANSWER KEY PAGE 151 33

 TABLE OF
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PRACTISE

7 Follow the steps from the previous page to represent the function g(x) 5 210 ​ __
1
[ ]
​   ​  (​x 2 16) ​​ 2 10.
8
a) Write the coordinates of a closed endpoint.
b) Determine the length of the steps.
c) Are the closed endpoints on the left or the right?
d) Is the function increasing or decreasing?
e) Determine the height of the jumps.
f ) Graph the function.

ICT
In ICT Activity 1.2.1, you can observe
how changing a, b, h and k affects the
graph of a greatest integer function.
This book is the property of Dickson Joseph.

See this activity on portailsofad.com.

3. From the Graph to the Rule of a Greatest Integer Function

You can now graph a greatest integer function expressed in standard form, using its rule. Conversely, you can

© SOFAD / All Rights Reserved.

also determine the rule for this type of function from its graph.

8 To the right is the graph of a greatest integer function.

y
a) Refer to the length and direction of the steps to 6
determine the values of parameters a and b.
5

4
b) Using the position of one of the steps, determine
3
possible values for parameters h and k.
2

1
c) What is the function rule?

0 1 2 3 4 5 6 x

34 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

REMEMBER

From the Graph to the Rule of a Greatest Integer Function

Follow the steps below to determine the rule for a greatest integer function in standard form:
f(x) 5 ab(x − h) 1 k.

Example:
Cost of parking Step Example
as a function of

SITUATION 1.2
ACQUISITION A
parking time 1) Determine possible values for (20, 1)
Cost
($) parameters h and k by selecting a
4 closed endpoint on the graph.
3 2) Determine |a| from the height of the 1
2 jumps.
3) Determine |b| from the inverse of the 1
1
length of a step. 20
0 20 40 60 80 100
4) Determine the sign of b from the According to the direction
direction of the closed endpoints. of the endpoints, b is
Time
(min) negative.
1
The rule for this function Therefore, b 5 2 .
20
is therefore 5) Determine the sign of a by noting Since the function is
[ 1
f(x) 5 2​​ 2___ ]
​   ​  (​x 2 20) ​​ 1 1.
20
whether the function is increasing or
decreasing and by taking into
increasing and parameter
b is negative, parameter a
account the sign of parameter b. will also be negative.
Therefore, a 5 21.

CAUTION!
An infinite number of rules may be associated with the graph of a greatest integer function. Parameters a
and b remain constant throughout, but this is not the case for parameters h and k. Each closed endpoint of
a step in the graph may be considered a possible value for parameters h and k.

PRACTISE
Late fee due as a
© SOFAD / All Rights Reserved.

function of number
9 The graph to the right illustrates the cost of the late fees
Amount of
of days late
Marie has to pay when she does not return video games late fee
($) 11
to the rental store in time. Determine the function rule 10
and evaluate the amount due if Marie returns a game 9
2 days late. 8
7
The rule: 6
5
The amount of the late fee:
4
3
2
1

0 1 2 3 4 5 6
Number of days late
(days)

ANSWER KEY PAGE 151 35

 TABLE OF
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CONTENTS

4. Nearest Integer Functions

4.1 The Nearest Integer Function for Positive Numbers
Rounding off a number means giving it an approximate value. Numbers can be rounded up, down or to the
nearest integer. This section focuses on the last of these types of rounding.

10 The table below shows the real temperatures recorded at a weather station and these same
temperatures as they would appear if rounded down or to the nearest integer.

Temperatures recorded (°C) 3.00 3.45 3.50 3.60 3.95

Rounded down 3 3 3 3 3
Rounded to the nearest integer 3 3 4 4 4

Rounding down corresponds to the greatest integer function f(x) 5 [x] you studied previously. By altering
the parameters slightly, you can find the function for rounding to the nearest integer.
a) In the example of recorded temperatures above, at what point does the nearest integer function no
longer yield the same value as the greatest integer function?
This book is the property of Dickson Joseph.

b) Is this true for all integers?

11 On the Cartesian plane below, graph the greatest integer function f(x) 5 [x] and show what the graph for
the nearest integer function should look like.

© SOFAD / All Rights Reserved.

36 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

12 In the graph on the previous page, you may have noticed that the same step pattern exists in both
functions. In fact, the nearest integer function produces jumps that begin 0.5 unit before those produced
by the greatest integer function. Using this information, it is possible to find the rule for the nearest
integer function by adjusting the greatest integer function rule.
You learned in the previous section that a horizontal translation to the left is produced by decreasing
parameter h. By decreasing parameter h by 0.5, the nearest integer function will therefore undergo a
horizontal translation to the left.
The resulting function rule is f(x) 5 x 1 0.5.

SITUATION 1.2
ACQUISITION A
a) Check whether the function seems to round any positive number to the nearest integer.

b) Although parameter h was decreased by 0.5 (h 5 20.5), the rule shows an addition of 0.5. Explain why.

4.2 The Nearest Integer Function for Negative Numbers

What happens when you round negative numbers to the nearest integer? Is the rule the same?
Answer the following questions to see what happens in this case.

13 What are the resulting values when you round the

following numbers to the nearest integer? CAUTION!
1) 23.4: 2) 23.5: 3) 23.6:
The mathematical convention for rounding
negative numbers to the nearest integer is not
14 Use the function f(x) 5 x 1 0.5 to evaluate the same as the one for positive numbers.
the following expressions. When a negative number is halfway between
1) f(23.4) 5 two integers, its value is rounded down to the
integer below.
2) f(23.5) 5
© SOFAD / All Rights Reserved.

To determine the rounded value of any

3) f(23.6) 5 number—positive or negative—that is halfway
between two integers, always choose the integer
that is further away from 0.
As you can see, the rule f(x) 5 x 1 0.5 does not
work with all negative numbers; you need to find
Example:
another way to round negative numbers to the
nearest integer. This is the task that awaits you in The rounded value of 21.5 is 22, just as the
the situational problem “Measuring the Temperature.” rounded value of 1.5 is 2.

23 2 21.5
2 1
2 1 1.5 2 3 R

ANSWER KEY PAGE 152 37

 TABLE OF
ANSWER KEY
CONTENTS

SOLUTION
You can now solve Situational Problem 1.2.
GREATEST INTEGER FUNCTION f(x) 5 a[b(x 2 h)] 1 k
SITUATION 1.2 NEAREST INTEGER FUNCTIONS

Measuring the Temperature SP 1.2

You must determine the function rule for A weather website shows the current
temperature in a few towns in the

rounding a number to the nearest integer so that Laurentians and the surrounding
areas. All of Québec’s weather stations
measure temperature precisely, to the

the Mont-Tremblant ski resort can automate its hundredth of a degree. However, tem-
peratures are usually broadcast to the
public as whole numbers.

information process. You must also determine

TASK

the interval containing the real temperature at

Mont-Tremblant when the weather report says it
is 0 °C outside. Show your work and support your
answer with graphs. These numbers are automatically rounded off by data processing software, without any human intervention.

© SOFAD / All Rights Reserved.

All temperatures measured across Québec are thus processed in a fraction of a second, using a real function.
It is important to understand how temperatures are rounded off in order to interpret them properly. For
example, as you certainly know, the freezing point of water is 0 °C; if the temperature rises above this point,
snow and ice melt. For the people who run the ski resort at Mont-Tremblant, it is important to know the real
temperature so they can plan which slopes should be open or shut and inform skiers. In this learning
situation, you need to provide Mont-Tremblant’s operators with a mathematical tool for making sound
decisions based on the temperature.

You must determine the function rule for rounding a number to the nearest integer so that the

TASK
Mont-Tremblant ski resort can automate its information process. You must also determine the
This book is the property of Dickson Joseph.

interval containing the real temperature at Mont-Tremblant when the weather report says it is
0 °C outside. Show your work and support your answer with graphs.

26 CHAPTER 1 – Step Functions and Greatest Integer Functions

Summary of the Facts SITUATIONAL PROBLEM FROM PAGE 26

• The freezing point of water is 0 °C.

• Above 0 °C, snow and ice begin to melt.
• Temperatures in weather reports are rounded to the nearest integer.

The Function for Rounding Off the Temperature

a) The rule for determining the rounded temperature within an interval [0, 1∞ °C:

© SOFAD / All Rights Reserved.

b) The graph of the function for rounding a negative temperature to the nearest integer:

38 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

c) The rule for determining the rounded temperature within an interval 2∞, 0 °C:

SITUATION 1.2
SOLUTION
d) The rule for the whole domain:
(Fill in the blanks.)

f(x) 5
{ , if x  ]2∞, 0[
, if x  [0, 1∞[

e) The graph for the whole domain:

(Show both rules on the same Cartesian plane, taking into account the domain of each rule.)
© SOFAD / All Rights Reserved.

The Interval

Answer:

ANSWER KEY PAGE 152 39

 TABLE OF
ANSWER KEY
CONTENTS

Mathematical knowledge
ACQUISITION B targeted:
•• writing a greatest
integer function from
a table of values
1. From the Table of Values to the •• interpreting the
properties of a greatest
Rule of a Greatest Integer Function integer function
f(x) 5 a[b(x 2 h)] 1 k
You have seen the relationship between the rule and the graph
of a greatest integer function several times in this chapter.
Now you will explore the relationship between the table of values
and the rule for a function of the form f(x) 5 ab(x – h) 1 k.

1 Consider the table of values to the right. Employee’s salary as

a function of seniority
a) What are the values of parameters a, b, h and k?
Seniority Salary
a5 b5 h5 k5 (years) (thousands of $)
b) Write the function rule. [0, 2[ 35
[2, 4[ 38
This book is the property of Dickson Joseph.

[4, 6[ 41
[6, 8[ 44
[8, 10[ 47

REMEMBER

From the Table of Values to the Rule of a Greatest Integer Function

The procedure for determining the rule for a greatest integer function in its standard form,
f(x) 5 ab(x – h) 1 k, from a table of values is similar to the steps you followed when starting from a
graph: you need to find the values of parameters a, b, h and k. However, these parameters look different
in the table of values and may not be so easy to identify. Simply follow these steps to find the rule.
1) Determine possible values of parameters h and k by selecting a closed endpoint on the graph.

© SOFAD / All Rights Reserved.

2) Determine |a| from the height of the jumps.
3) Determine |b| from the inverse of the length of a step.
4) Determine the sign of b by noting which side of the interval is closed. If the interval is closed on the
left, b is positive; if the interval is closed on the right, b is negative.
5) Determine the sign of a by noting whether the function is increasing or decreasing and by taking
into account the sign of parameter b.

40 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Example:
x f(x)
[25, 21[ 12
difference of 6
[21, 3[ 6
difference of 6
[3, 7[ 0
difference of 6
[7, 11[ 26

1) The endpoint (3, 0) is closed. Therefore, two possible values of parameters h and k are h 5 3

SITUATION 1.2
ACQUISITION B
and k 5 0.
2) The difference between two consecutive values of the dependent variable is 6 (12 2 6 5 6),
so the jumps have a height of 6 units and |a| 5 6.
3) The difference between the endpoints of an interval is 4 (21 2 (25) 5 4), so the steps are 4 units
1
long and |​b| 5  ​ __.​​
4
4) The intervals are closed on the left, so the closed endpoints are at the beginning of the steps
and parameter b is positive.
5) When the values for x increase, the values for y decrease, so the function is decreasing.
A decreasing function means that a and b have opposite signs. Since parameter b is positive,
parameter a must be negative.

[
Consequently, the function rule is f(x) 5 26​​ __
1
]
​   ​ (​x 2 3) .​​
4

PRACTISE

2 The table of values below represents the cost of international calls made by Jade with her prepaid card.
Cost as a function of length of call
Length of call Cost of call
(min) ($)
]0, 1] 0.055
© SOFAD / All Rights Reserved.

]1, 2] 0.11
]2, 3] 0.165
]3, 4] 0.22

a) Determine the function rule.

ANSWER KEY PAGE 153 41

 TABLE OF
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CONTENTS

b) How long can Jade talk to her aunt if she has $2.53 left on her card?

c) How much would a call of 26.5 min cost?

2. The Properties of a Greatest Integer Function

This book is the property of Dickson Joseph.

To complete your study of greatest integer functions, answer the

following questions to learn more about their properties.
Cost of renting a
car as a function
3 To the right is a graph illustrating the cost of renting a car
Cost
of number of days
on a trip abroad. For customers interested in rental of up ($)
240
to two weeks (14 days), the company offers the following
package: $70.00 for the first two days, plus $50.00 for each 200

additional two-day period. 160

120
Answer each of the following questions about the package
and indicate which function property it refers to. 80

40
a) How long could you rent a car for?

© SOFAD / All Rights Reserved.

0 2 4 6 8 10
Number
of days

b) What are the amounts you could have to pay?

c) What is the minimum amount you could pay?

d) What is the maximum amount you could pay?

42 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

e) Does the cost increase or decrease?

CAUTION!
To answer the questions above, you had to study the function carefully. Note that the context can influence certain
properties of the function. In this case, the sign of the function is always positive because it is always true that
f(x)  0 (you cannot pay a negative amount to rent a car). Also, since it is impossible to rent a car for 0 days, there

SITUATION 1.2
ACQUISITION B
is no y-intercept. Finally, since the minimum amount a customer may pay is $70.00, the function has no zero.

REMEMBER

Function Properties Applied to Cost of sending a parcel

the Greatest Integer Function Cost
depending on its weight
($)
To examine the various properties of a 25
24
function, consider this example of the function 23
g(x) 5 3[0.5x] 1 15, which represents the cost 22
of mailing a parcel as a function of its weight 21
20
in kilograms. Range
19
18
y-intercept
17
16
15
no zero
0 2 4 6 8 10
Weight
(kg)
Domain

Property Definition Example

Domain The set of values that the independent variable may [0, 1∞[ kg
have
Range (codomain) The set of values that the dependent variable may have ${15, 18, 21, 24, …}
© SOFAD / All Rights Reserved.

Zeros of the function The values for which f(x) 5 0 ∅ (no zero)
(x-intercepts)
Initial value The value of f(0) In specific contexts, it is often called the $15
(y-intercept) start value. In a graph, it is called the y-intercept.
Sign The sign of a function may be positive or negative. It is positive over
expressed by the intervals of the domain of the function x  [0, 1∞[ kg
where f(x)  0 if the function is positive and where f(x) 
0 if the function is negative.
Change The direction of change indicates whether the function Increasing over the entire
is increasing or decreasing. domain
Extrema The minimum is the smallest of the values of the No maximum
(minimum or maximum) dependent variable. The maximum is the largest value. Minimum: $15

ANSWER KEY PAGE 153 43

 TABLE OF
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CONTENTS

PRACTISE

4 Determine the properties of the greatest integer function illustrated below.

a) Domain:
y

5
b) Codomain:
4
3 c) x-intercept:
2
1 d) y-intercept:

5 24 23 22 21
2 1 2 3 4 5 x e) Sign:
22

23 f ) Direction:
24

25 g) Extrema:
h) What is the function rule?

5 An event planner has an idea for a cystic fibrosis fundraiser. The event, called “Every Breath
Counts,” is taking place at the local pool. The city offers to donate $1.00 for every 10 seconds
each participant can hold his or her breath under water, up to a maximum of 1 min.
a) Graph this situation.
This book is the property of Dickson Joseph.

© SOFAD / All Rights Reserved.

b) Determine the properties of the function.
1) Domain:
2) Codomain:
3) x-intercepts:
4) y-intercept:
5) Sign:
6) Direction:
7) Extrema:
8) What is the function rule?

Your study of step functions and, more specifically, of the greatest integer function ends here.
Consolidate your newly acquired knowledge by doing the exercises in the next section.

44 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

CONSOLIDATION

1 [
Let the function g(x) 5 2​​ __
1
]
​   ​  (​x − 2) ​​ − 1.
3
a) Is the function increasing or decreasing? Explain your answer.

SITUATION 1.2
CONSOLIDATION
b) What is the direction of the steps in the graph of the function?
Circle the correct form of step and explain your answer.

c) What is the length of the steps?

d) What is the height of the jumps?
e) Graph the function.

2 Let the function f(x) 5 4 2 2[3 1 0.5x].

a) Write the function rule in the form f(x) 5 ab(x 2 h) 1 k.
© SOFAD / All Rights Reserved.

b) What are the values of parameters a, b, h and k?

a5 b5 h5 k5
c) Graph the function f.

ANSWER KEY PAGE 154 45

 TABLE OF
ANSWER KEY
CONTENTS

d) What is the range of f?

e) What is the direction of change for the function?
f) Refer to the graph to determine the zero set of f.

3 Find the values of parameters a, b, h and k, then the rule, for each of the functions represented below.
a) b) x g(x)
y

6 ]25, 21] 5
5 ]21, 3] 2.5
4
]3, 7] 0
3
2
]7, 11] 22.5

1 ]11, 15] 25

2 6 5
2 4
2 2 3 2
2 1
2 1 2 3 4 5 6 x

2
2

3
2

4
2

5
2

6
2
This book is the property of Dickson Joseph.

a5 b5 a5 b5
h5 k5 h5 k5
f(x) 5 g(x) 5

4 The graph to the right represents the function f(x) 5 x.

y
5
4
3
2
1

5
2 24 2 3 2
2 1
2 1 2 3 4 5 x

© SOFAD / All Rights Reserved.

2
2

3
2

4
2

5
2

a) Graph the function h(x) 5 2x 1 3 2 1.

46 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

b) What transformations did the initial graph undergo to produce the final graph?

5 Determine the rule for the greatest integer function represented x y

by the table of values to the right.
[210, 25[ 2
[25, 0[ 4

SITUATION 1.2
CONSOLIDATION
[0, 5[ 6
[5, 10[ 8

The rule:

6 The zero set of a greatest integer function f is the interval [12, 17[. The function is increasing, and the
image of 17 under this function is 8.
a) Determine the function rule.
(Suggestion: Sketch the graph on a piece of scrap paper.)
b) What is the y-intercept of the function?

7 Refer to the following graph to answer the questions below.

a) Determine the function rule.
y

10
8
6
4
2

2 28 224 220 216 212 28 422

2 4 8 12 16 20 24 28 x
b) What effect would you observe on the graph
4
if parameter a were increased by 2 units?
2

6
2

8
2
© SOFAD / All Rights Reserved.

2 10

c) What effect would you observe on the graph

if parameter h were decreased by 3 units?

ANSWER KEY PAGE 155 47

 TABLE OF
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CONTENTS

8 Consider the following diagram.

Add Divide Apply Multiply

x
5 by 10 integer part by 10

a) Determine the results of applying the operations in the diagram to the following values of x.
1) x 5 98: 2) x 5 123: 3) x 5 235:
b) The diagram represents a function. Describe the purpose of this function.

c) Write the function rule.

9 Oliver borrowed $500 from his father to buy a new laptop. Although Oliver will not pay interest, he has
agreed to pay his father back $50 on the 7th day of each week.
Let D(x), where D is Oliver’s debt in dollars and x, the number of days after the loan.
a) Fill in a table of values and a graph to represent the function D.
This book is the property of Dickson Joseph.

© SOFAD / All Rights Reserved.

b) What is the function rule?

c) How much money will Oliver owe his father

after 30 days? Justify your answer.

48 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

10 The number of workers in a daycare centre depends on the number of children registered.
The rule for the function that determines how many workers are needed is as follows: W(x) 5 ​​ __
x
[]
​  ​ ​​1 2,
7
where x represents the number of children registered and W(x), the number of workers needed.
a) What does parameter k represent in this context?

b) What does parameter b represent in this context?

SITUATION 1.2
CONSOLIDATION
c) If 43 children are registered for this coming September, how many workers are needed to run the
daycare centre?

d) If 14 workers are employed, how many children will be attending the centre?

11 Matthew organizes bus trips to the local casino. He uses the company Locabus, which charges $72.99 for
the first group of 40 passengers, plus $50.00 for each additional group of 40.
a) Determine the rule representing this situation.

b) Matthew believes that 239 people will sign up for the next trip. How much money will he have to pay
Locabus?
© SOFAD / All Rights Reserved.

c) In the end, the bill for renting the buses comes to $372.99. How many people signed up?

ANSWER KEY PAGE 156 49

 TABLE OF
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CONTENTS

This is a summary of
KNOWLEDGE SUMMARY you need to REMEMBER
what
in the missing informa
. Fill
tion.

Step Functions
A step function is a function that is constant on each of its defining intervals and that jumps from one interval
to the next as the independent variable changes.
The are the endpoints of the intervals where the function varies abruptly.

As a result, the graph of the function is made up exclusively of horizontal segments called steps.
A(n) () at the end of a step means that it is included in the graph of the function.
A(n) () means the opposite. The image of a critical value always corresponds to the
y-coordinate of the closed circle.

Example:

Graph of a step function Table of values • The critical values of the

This book is the property of Dickson Joseph.

y function are .
5 • The closed endpoints
4 x f(x)
3 associated with these critical
2 [23, 21[ 3 values mean that f(21) 5 22
1
[21, 1[ and f(1) 5 2.
25 24 23 22 21 1 2 3 4 5 x
22
[1, 4]
23

24

25

Greatest Integer Function f(x) 5 [x]

Greatest integer functions are specific cases of functions.

© SOFAD / All Rights Reserved.

The integer part of a number, written x, is the greatest integer less than or equal to that number.

In the graph of a greatest integer function, all the steps are of the same length, and the jumps between
consecutive steps are equal in height. Step functions are also called staircase functions because of their
obvious resemblance to a staircase.

The rule for this greatest integer function is written as follows: .

Examples:
__ __
f(2.3) 5 2.3 5 2 ; f(​√
​2 ​​) 5 ​√
​2 ​​ 5 1 ; f(213.5) 5 213.5 5 214

50 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Greatest Integer Function f(x) 5 a[bx]

In a greatest integer function of the form f(x) 5 abx:
• The first closed endpoint is at (0, 0).
• Parameter a determines the height of a jump; this height is equal to .
• Parameter b determines the length of a step; this length is equal to .

Example:

CHAPTER 1
KNOWLEDGE
SUMMARY
Graph of a greatest integer function Table of values Height of one jump: 4

y Length of one step: 3

8
Value of parameter a: 4
7 x f(x)
1
Value of parameter b: __
6
​​   ​​
5 [26, 23[ 28 3
4
3
2 Jump [23, 0[ 24 Rule:
Closed
endpoint 1
[0, 3[
27 26 25 24 23 22 21 1 2 3 4 5 6 7 8 9 x
22

23
[3, 6[
24

25 [6, 9[
26
Step 27

28

The Values of a Greatest Integer Function

To determine a value of x when f(x) is known, follow these steps:
1) Isolate the integer part.
2) Determine the interval that satisfies the integer part. This interval always takes the form z, z 1 1,
where z is an integer.
3) Isolate x in the expression contained in the integer part.
© SOFAD / All Rights Reserved.

To determine a value of f(x) when x is known, simply replace x in the rule with its value and follow the order of
operations. In this case, the integer part symbol has the same priority as brackets.

Example:
Let the function f(x) 5 230.25x.
If the value of x is 11, If the value of f(x) is 9, If the value of f(x) is 27, then:
then: then: 27 5 230.25x
f(11) 5 230.25 × 11 9 5 230.25x _
2. 3 5 0.25x
f(11) 5 232.75 23 5 0.25x
Since no _number has an integer
f(11) 5 23 3 2 5 23  0.25x , 22 part of 2.​​3​​, there is no
.

KNOWLEDGE SUMMARY PAGE 122 51

 TABLE OF
ANSWER KEY
CONTENTS

Greatest Integer Function f(x) 5 a[b(x 2 h)] 1 k

The Effect of Changing Parameters a, b, h and k
• Reversing the sign of parameter a produces a reflection over the -axis.
• Reversing the sign of parameter b produces a reflection over the -axis.
y y

1 1 If parameter b If parameter b
is positive is negative
1 x 1 x

Effect of reversing the Effect of reversing the

sign of parameter a sign of parameter b

• Increasing |a| increases the height of the jump. The function is said to undergo a vertical stretch.
Decreasing |a| has the opposite effect: a vertical compression.
This book is the property of Dickson Joseph.

• Increasing |b| decreases the length of the step. The function is said to undergo a horizontal compression.
Decreasing |b| has the opposite effect: a horizontal stretch.
• Increasing parameter h results in a(n) translation.
Decreasing h has the opposite effect: a horizontal translation to the left.
• Increasing parameter k results in an upward vertical translation. Decreasing k has the opposite
effect: a(n) .

Example:
In the function f(x) 5 (x 2 4) 1 5, the f(x) = [x] function undergoes a horizontal translation of
and a vertical translation of .
Consequently, there is a closed endpoint at (4, 5).

© SOFAD / All Rights Reserved.

52 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Graphing a f(x) 5 a[b(x 2 h)] 1 k Function

To graph a greatest integer function, you must determine the following information from the parameters of
the standard form:
• Determine the coordinates of a closed endpoint: (h, k)
1
• Determine the length of one : ______
​​   ​​.
|​ b ​|
• Determine whether the closed endpoints are on the left (b . 0) or the right (b , 0).
• Determine whether the function is increasing (a and b have the same sign) or
(a and b have opposite signs).

CHAPTER 1
KNOWLEDGE
SUMMARY
• Determine the height of the : |a|.

From the Graph to the Rule of a Greatest Integer Function

To determine the rule for a greatest integer function in its standard form, f(x) 5 ab(x − h) 1 k, you must find
the values of parameters a, b, h and k by following these steps:
1) Determine possible values of parameters h and k by selecting a(n) on the graph.
2) Determine |a| from the height of the .

3) Determine |b| from the inverse of the length of a(n) .

4) Determine the sign of b from the direction of the closed endpoints.
5) Determine the sign of a by noting whether the function is and by taking into
account the sign of parameter b.

Example:
Find the corresponding rule for the following graph.
y
24
1) Possible values of parameters h and k are h 5 22 and k 5 0.
16
2) |a| 5 4
1
3) |b| 5 __
8
​​   ​​
2
© SOFAD / All Rights Reserved.

4) According to the direction of the endpoints, b is positive. 2 8 6

2 4
2 4 8 12 x
8
2

Therefore, b 5 .
16
2
5) Since the function is decreasing and parameter b is positive,
24
parameter a will be negative. Therefore, a 5 . 2

Consequently, the function rule is f(x) 5 .

KNOWLEDGE SUMMARY PAGE 124 53

 TABLE OF
ANSWER KEY
CONTENTS

From the Table of Values to the Rule of a Greatest Integer Function

Follow these steps to determine the rule for a greatest integer function of the form f(x) 5 ab(x 2 h) 1 k,
using a table of values.

1) Determine possible values of parameters h and k. In the table of values, parameter h is the value
of the closed side of the interval and parameter k is the value of the associated .

2) Determine |a| from the . In the table of values, the height of the
jumps is the difference between two consecutive values of the dependent variable.

3) Determine |b| from the inverse of the length of a step. In the table of values, the length of the steps
is the of each interval.

4) Determine the sign of b by noting .

If the interval is closed on the left, b is positive; if the interval is closed on the right, b is negative.

5) Determine the sign of a by noting whether the function is and by taking

into account the sign of parameter b.
This book is the property of Dickson Joseph.

Example:
The steps below show how to determine the corresponding rule for the following table of values.

x y
[26, 21[ 12
difference of 3
[21, 4[ 9
difference of 3
[4, 9[
difference of 3
[9, 14[

1) The endpoint (h, k) 5 (4, 6) is closed because the interval [4, 9[ is closed at 4 and the value of the
y-coordinate at this point is 6. Therefore, two possible values of parameters h and k are h 5 4 and k 5 6.
2) The difference between two consecutive values of the dependent variable is 3 (12 2 9 5 3),

© SOFAD / All Rights Reserved.

so the steps have a height of 3 units and .
3) The difference between the endpoints of the intervals is 5 (21 2 (26) 5 5), so the steps
are 5 units long and |b| 5 .
4) The intervals are closed on the left, so the closed endpoints are at the beginning of the steps and
parameter b is positive.
5) When the values for x increase, the values for y decrease, so the function is decreasing. A decreasing
function means that a and b have opposite signs. Since parameter b is positive, parameter a must be
negative.
The function rule is therefore .

54 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Function Properties Applied to the Greatest Integer Function

Example:
y
The graph to the right represents the f(x)  2[2x]
6
function f(x) 5 2[2x].
4
Images

CHAPTER 1
KNOWLEDGE
Zeros

SUMMARY
3 2 1
 1 2 3 x

2
 y-intercept

4


The function is negative. The function is positive.

Domain

Property Definition Example

Domain The domain of a function is the set of values that Set of real numbers: R
the may have.

Range The range (or codomain) of a function is the set {… 26, 24, 22, 0, 2, 4, 6 …}
(codomain)
of values that the may have.

Zeros of the function The zeros of the function are the values for which
(x-intercepts)
. There may be none, only one
or several.

y-intercept The initial value is the value of f(0). In specific contexts, it

is often called the start value. In a graph, it is called the
y-intercept. Out of context, the term
y-intercept is more common.
© SOFAD / All Rights Reserved.

Sign The sign of a function may be positive or negative.

It is expressed by the intervals of the domain of the Negative over
function where f(x)  0 if the function is positive and
where f(x)  0 if the function is negative. Positive over

Change The direction of change indicates whether the function

over
is .
the entire domain

Extrema
The extrema are values, if they exist.

KNOWLEDGE SUMMARY PAGE 126 55

 TABLE OF
ANSWER KEY
CONTENTS

INTEGRATION

1 In an office, 250-ml cups are used to take water from a water cooler.
a) How many of these cups would be needed to empty the cooler
if it contained the following amounts of water?
1) 900 ml
2) 7.2 L
3) 16 L
b) Explain your calculations step by step for determining the number
of cups needed depending on the amount of water in the cooler.

c) Let N(x), the number of cups needed if there are x litres of water in
This book is the property of Dickson Joseph.

the cooler. Write the rule for the function N(x).

d) Graph the function.

© SOFAD / All Rights Reserved.

e) How many litres of water are in the cooler if it takes 3 cups to empty it?

2 Using the information provided, determine the rule for the greatest
integer function described.
a) My domain is R. I am increasing, my y-intercept is 3 and my
x-intercepts are 21, 0.

56 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

b) My domain is R and my range is {…, 24, 22, 0, 2, 4, …}.

My x-intercepts are 7, 11. I am strictly positive over x , 7.

INTEGRATION
CHAPTER 1
c) My domain is R and my range is {…, 23, 21, 1, 3, …}.
I am strictly positive for x . 1 and strictly negative for x  1.
When 1 , x  3, the value of my dependent variable is 1.

3 Is the function f(x) 5 3 2 2[4x 2 10] increasing or decreasing? Justify your answer.

4 Truncating a number to the integer means removing its fractional part and keeping only its integer part.
Example: The truncation of 3.59 is 3, while that of 22.8 is 22.
Model this type of rounding by writing a greatest integer function rule for each of the following parts:
negative numbers and positive numbers.
© SOFAD / All Rights Reserved.

ANSWER KEY PAGE 157 57

 TABLE OF
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5 Find the function rule represented by each of the graphs below.

a) b)
y y

8 12

4 6

28 24 4 8 x 220 10
2 10 20 30 x
4
2 6
2

8
2 12
2

6 In a 400-m women’s freestyle race, the competitors swim 8 lengths of the pool (8 times 50 m). During the
race, good execution on the turn is crucial for a swimmer. There are 7 turns in all: one at the end of each
of the first 7 lengths.
How can you determine the number of turns a swimmer has left, using the distance she has already
covered in the 400-m race? For example, at the start, she still has 7 turns to do, but when she has swum
This book is the property of Dickson Joseph.

50 m, she has just done a turn, so there are only 6 left.

a) Graph the function.

© SOFAD / All Rights Reserved.

b) Determine the function rule.

c) What are the zeros of the function?

d) Interpret the zero set in this context. What do the zeros correspond to?

58 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
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7 An activity club rents out a school’s pool to offer swimming lessons. Below are the rates for renting the
pool. The school never rents out its pool for more than 120 min at a time.

Rental time (min) Rate

0 min to 30 min inclusive $65
Over 30 min to 60 min inclusive $105
Over 60 min to 90 min inclusive $145
Over 90 min to 120 min inclusive $185

INTEGRATION
CHAPTER 1
a) Is the function increasing or decreasing? Explain your answer.

b) What are the domain and the range of the function?

c) Graph this situation.

8 A financial security adviser is paid by commission. She receives a base salary of $250 per week plus a $95
commission for every $2000 in sales she makes during the workweek.
a) Determine the function rule.
© SOFAD / All Rights Reserved.

b) What value of sales does she need to make if she wants to be paid at least $950 per week?

ANSWER KEY PAGE 157 59

 TABLE OF
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CONTENTS

c) If she makes $36 879 in sales in one week, how much will she be paid?

9 At La Mauricie National Park, visitors can rent canoes or kayaks. The cost of renting a kayak is $14 for the
first 90 min plus $8 for each additional 90-min block, whether full or partial.
a) Determine the rule representing this situation.

b) How much would it cost to rent a kayak for 4 hours?

This book is the property of Dickson Joseph.

c) If a rental cost a total of $38, how long did the visitor rent the kayak?

© SOFAD / All Rights Reserved.

60 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

10 Alexis would like to switch Internet providers. Two companies have caught his attention.
Below are the details of their packages.
Cable Distribution
$30 per month for basic service including up to 20 gigabytes (GB) download.
If download exceeds 20 GB, an extra $5 will be charged for each additional 20 GB block
(complete or not).

Fibre-plus
$20 per month for basic service including up to 10 GB download.

INTEGRATION
CHAPTER 1
If download exceeds 10 GB, a $5 supplement will be charged for each additional 10 GB block
(complete or not).
What advice would you give Alexis to help him choose the right package?
Support your answer with mathematical arguments, including a graph.
© SOFAD / All Rights Reserved.

Your advice:

ANSWER KEY PAGE 158 61

 TABLE OF
ANSWER KEY
CONTENTS

LES
This book is the property of Dickson Joseph.

Timekeeping
A newspaper describes the results of a
Rubik’s cube competition.
As you can see in the table, the event seems to have been
RUBIK’S CUBE SOLVED IN UNDER 10 SEC
timed to the tenth of a second. However, the recorded times
A regional Rubik’s cube competition was do not precisely reflect the real times achieved by the
held yesterday at the Science Centre. The finalists. In fact, in international Rubik’s cube competitions,
day ended with the top four competitors times are often measured to the hundredth or even the
facing off in a spectacular final. The thousandth of a second. Real time can be subdivided as
winner, Oliver Little, astonished many times as you wish. For example, Oliver Little may have
onlookers by solving the cube in under solved the cube in 9.62 s or even 9.685 s. Both cases are
10 seconds. Almost as incredibly, two possible because the timer stopped at 9.6 s and therefore

© SOFAD / All Rights Reserved.

other finalists achieved identical times. did not reach 9.7 s.
Look at the table again. The same time is indicated for both
Finalist Best time (s) Waheb Jawad and Mike Cheung: 16.8 s. Does this mean that
Oliver Little 9.6 each of them solved the cube as quickly as the other?
Anne Terry 14.3 The action of the timekeeping device can be described
Waheb Jawad 16.8 using the diagram below.
Mike Cheung 16.8
Real time Action Recorded time
9.685 s of timer 9.6 s

Determine the possible real times of Waheb Jawad and Mike Cheung from their recorded times.
TASK

State whether you think each competitor solved the cube as quickly as the other. Justify your
answer by modelling the action of the timer with a function. Represent this function by its rule and
by another means (table of values or graph).

62 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
ANSWER KEY
CONTENTS

Representation of the Function Modelling the Action of the Timer

CHAPTER 1
LES
© SOFAD / All Rights Reserved.

Function rule:

Answer:
Evaluation by criterion
Cr. 1.1 A B C D E
Cr. 1.2 A B C D E
Cr. 2.1 A B C D E
Cr. 2.2 A B C D E
Cr. 2.3 A B C D E

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CHAPTER 2

Algebraic Expressions

Revealing the Secrets of

Paranormal Activity

S
ome people call themselves mentalists, psychics, mystics,
magicians, numerologists or clairvoyants and claim to have
paranormal powers which they attempt to prove exist.
This book is the property of Dickson Joseph.

Few of these “gifts” remain unexplained and most are not real.
For example, magicians claim to possess extraordinary powers, but
in fact use tricks that only they know. Often, a trick is simply a very
clearly defined procedure that includes mathematical operations
which the magician has prepared ahead of time. Since the
magicians we meet in this chapter will not be revealing their
secrets themselves, it will be up to you to see through their tricks
using tools you have already encountered and will get to know
better in this chapter: mathematical reasoning and the rules of
algebra. In this quest, you will discover special algebraic identities

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and use them to simplify expressions and factor polynomials.

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CHAPTER 2
SITUATION 2.1
RATIONAL EXPRESSIONS
PERFECT SQUARE TRINOMIALS
DIFFERENCE OF SQUARES
SP 2.1 – The Prodigy p. 66

SITUATION 2.2
MULTIPLYING TWO POLYNOMIALS
DIVIDING A POLYNOMIAL BY A BINOMIAL



SP 2.2 – The Telepath p. 86

KNOWLEDGE SUMMARY p. 104

INTEGRATION p. 108
LES
The Amateur Mentalists p. 112
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65
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LAWEXPRESSIONS
RATIONAL OF COSINES
TRIGONOMETRIC
PERFECTFORMULAE FOR AREA
SQUARE TRINOMIALS
SITUATION 2.1 HERON'S OF
DIFFERENCE FORMULA*
SQUARES

The Prodigy SP 2.1

During a variety show, a performer

appears on stage and introduces himself
as a mental calculation prodigy. He claims
to be able to solve complex mathematical
problems in a fraction of a second without
the help of a calculator or even a pencil.
He challenges anyone in the room to solve
This book is the property of Dickson Joseph.

a complex mathematical expression faster

than him after replacing the variable with
any number.

The prodigy asks an audience member to pick a number, which he will substitute
into the expression on the screen. She picks number 7. The prodigy answers: _​​  7 ​​.
9

The prodigy then invites other audience members to give him different
numbers. Someone shouts out “Five!” Someone else says “Three!” In next to no
7 5
time, the prodigy answers: __
​​  ,​​ __
​​  .​​
5 3
TASK

You must demonstrate that the performer does not really have a prodigious talent
and identify the mathematical procedure he uses to solve the equation so quickly.

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EXPLORATION
The questions in this exploration activity will help you to analyze the sequence of operations in the
expression given by the prodigy. Using this analysis, you will be able to determine the numerical
value of the expression when the variable is replaced by a number.

1 Consider the expression proposed by the prodigy again. Replace the letter n by 7, then check
that the answer he gave is indeed correct.
n 1 1 ____
____ 1 2 (7) 1 1 1 2
​ ​1 ​ ​1 ______
​ ​5 ​_____​1 ​_____​1 ________
​ ​
n n 1 2 n(n 1 2) (7) (7) 1 2 (7)((7) 1 2)

Situation 2.1
exploration
REMINDER REFRESHER EXERCISES
PAGE 118, QUESTIONS 12 AND 13
Addition and Multiplication of Fractions
To add or subtract two or more fractions, you need to:

Procedure Example
1 2 __
__ 3 4
1. Give the fractions the same denominator. ​ ​1 __
​ ​5 ​ ​1 __
​​
2 3 6 6
314
____
2. Add the numerators. ​ ​
6
7
__
3. Simplify the outcome, if applicable. ​​
6

To multiply two or more fractions, you need to:

Procedure Example
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1 2 _____
__ 132
1. Multiply the numerators, then multiply the denominators. ​ ​3 __
​ ​5 ​ ​
2 3 233
2 1
__
2. Simplify the outcome, if applicable. ​ ​5 __
​​
6 3

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2 Use the same steps for different Solution to The Prodigy

values of n and fill in the table. Number chosen Resulting fraction Simplest form
a) When n 5 1.
1

5
__ 5
__
3 ​​ ​​
3 3
b) When n 5 2.
4

7
__ 7
__
5 ​​ ​​
5 5

9
__ 9
__
7 ​​ ​​
7 7

3 What link can be made between the denominators of the fractions in the initial answer
This book is the property of Dickson Joseph.

(before simplification) and the numbers chosen at the start?

4 Is there a link between the numerators of the fractions obtained in the answer and the numbers
chosen at the start?

STRATEGY Proof based on reasoning

Verifying a statement using several numerical examples is a good way to validate a conjecture, but it does not
constitute a mathematical proof. To prove that a statement is true beyond any doubt, you must use reasoning

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that applies to all possible situations. In this case, algebra must be used.

5 Formulate a conjecture regarding the link between the outcome and the number chosen at the start.

You have observed that it is possible to replace a variable with a value in an algebraic expression to produce
an equation. However, to prove that the equation is always true, algebraic expressions must be manipulated.
The acquisition activity that follows will help you to prove, in a formal way and using using algebraic
reasoning, that an identity is always true.

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ACQUISITION A Mathematical knowledge

targeted:
•• recognizing rational
expressions
1. Rational Expressions •• determining restrictions
•• adding and subtracting
In the next section, you will learn about rational expressions, and rational expressions
the restrictions that go with them. These expressions are also called
•• factoring perfect square
“algebraic fractions.” They resemble the numerical fractions you already trinomials
know. Firstly, they are written with a numerator and a denominator. •• simplifying algebraic
Secondly, as you will see in the following sections, all the properties and fractions by factoring
arithmetic processes associated with numerical fractions also apply to
rational expressions.

1 Simplify the following expressions.

12x
a) ____ 24​a​​
b) _____ 14​y​3​
c) ____
​ ​ ​ 2​ ​ 3​
3 8​a​ ​ 7​y​ ​

Situation 2.1
Acquisition A
REMINDER REFRESHER EXERCISES
PAGE 118, QUESTIONS 14 TO 17
Simplifying Fractions
To simplify a numerical fraction the common factors of the numerator and denominator must be
eliminated. This is also true for algebraic fractions.

The Laws of Exponents

The algebraic manipulation of expressions containing exponents is subject to certain laws.

a0 5 1 (am)n 5 amn ​a​m​

___ 1
​ n ​​5 am 2 n a2m 5 ___
​ m​
​a​ ​ ​a​ ​
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a1 5 a am 3 an 5 am 1 n

Example:
Exponents are added. Exponents are subtracted.
24​y​8​​z​4​
_______
5x2 3 22x3 5 210x5 ​ 3 3 ​5 8y5z1
3​y​ ​​z​ ​
Coefficients are multiplied. Coefficients are divided.

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2 Use algebraic fractions to translate

the following statements: CAUTION!
a) a number divided by 2 In these translated fractions, the numbers cannot be
assigned just any value because no number can be
divided by 0.
b) a number plus 1, divided by the double
• Since 0 is the absorbing element of multiplication,
of that number
any number multiplied by 0 will always give a
result of 0.
c) a number plus 1 divided by the double • The algebraic expression a 3 0 5 b, where b  0,
of that number is an impossible equation, as the value of the
product can only be 0.
b
• Likewise, the equivalent equation __
​ ​​5 a, where
d) a number divided by 2 more than 0
b  0, is also impossible.
that number
Therefore, the variables of a rational expression may
take any value as long as the denominator is not zero.

REMEMBER
This book is the property of Dickson Joseph.

Rational Expressions
P
A rational expression can be written in the form __
​ ,​​ where P and Q are polynomials, but where Q  0.
Q

Restrictions
A rational expression is properly Example:
defined only when the divisor is 3
Consider the rational expression _____
​ .​​
different from 0. The values that the 2x 1 5
variables of this expression may not To find the restrictions, you must solve 2x 1 5  0:
take are called the restrictions.
2x 1 5 2 5  25 (subtract 5 on each side)
Before exploring operations on the
2x 25
___
rational expressions, it is essential to ​ ​​ ___
​ ​​ (divide by 2 on each side)
2 2
name these restrictions. 25 25
x  ​___​​ (therefore, x cannot take the value ___
​ )​​
2 2

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PRACTISE

3 Determine, as applicable, the restrictions for the following expressions, then simplify where possible.
14x
a) ____ 22x
b) ​____ 3​y​​
c) _______
​ ​ ​ 2 ​​ ​ ​
5 12x 6y(x 1 2)

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2. Adding and Subtracting Rational Expressions

In Situational Problem 2.1, what appears on the screen is a sum of rational expressions similar to this one:
3
____ 2a 2 1 ______ 1
​ ​1 _____
​ ​2 ​ ,​​ where a  0 and a  1
a21 a a(a 2 1)
To solve the “enigma,” you need to learn how to manipulate such expressions. In this section, you will discover
how to add and subtract rational expressions.

4 Add the following fractions.

1 1
a) __ 2 3
b) __ 1
c) ___ 5
​ ​1 __
​​ ​ ​1 __
​​ ​ ​1 __
​​
2 6 5 4 2x 6

5 Subtract the following fractions.

Situation 2.1
7 3 3 2a 1

Acquisition A
___
​ ​2 __
a) ​ ​ ​​ b) ​​___​2 ___
​ ​2 ___
​ ​
12 4 5a 3 2a

Tip
When adding or subtracting algebraic fractions, you must apply the same procedure as you
would apply to numerical fractions.

Numerical fraction Algebraic fraction

1 7 1 3 5 ___ 7 Give the fractions the 2
__ 3 2 3 2 ___ 3
__
​ ​1 ___
​ ​5 _____
​ ​1 ​ ​ ​ ​1 ___
​ ​5 _____
​ ​1 ​ ​
3 15 3 3 5 15 same denominator. a 2a a 3 2 2a
Name the restrictions. Restriction: a  0
5
___ 7 12 4 Add or subtract the 4 3 7
​ ​1 ___
​ ​5 ___
​ ​5 __ ___
​ ​1 ___
​ ​5 ___
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​​ ​ ,​​ where a  0
15 15 15 5 transformed fractions. 2a 2a 2a

Simplify, if applicable. Apply restrictions.

It is important to remember to name the restrictions when manipulating rational expressions.

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REMEMBER

Adding and Subtracting Rational Expressions

To add and subtract rational expressions, you must:
• determine the common denominator;
• transform all fractions into equivalent fractions with the same common denominator;
• add or subtract the numerators, then simplify the similar terms in the numerator position;
• simplify the expression, if applicable.

NOTE: Remember to name the restrictions at the start.

Example:
3 2a 2 1 ______1
Below is how to solve ____
​ ​1 _____
​ ​2 ​ .​​
a21 a a(a 2 1)

Explanations Example
3
____ 2a 2 1 ______1
The common denominator is a(a 2 1). ​ ​1 _____
​ ​2 ​ ​
a21 a a(a 2 1)
This book is the property of Dickson Joseph.

Name the restriction(s). Restrictions: a  1 and a  0

Perform the necessary multiplications to get 33a (2a 2 1)(a 2 1) 1
5 ________
​ ​1 ​___________​2 ______
​ ​
the equivalent algebraic fractions. (a 2 1) 3 a a(a 2 1) a(a 2 1)
Keep the denominator in factor form to make 3a 2​a​2​2 2a 2 a 1 1 1
​5 ​______​1 ​___________​2 ​______​
future possible simplifications easier. a(a 2 1) a(a 2 1) a(a 2 1)
3a 1 2​a​2​2 2a 2 a 1 1 2 1
With a common denominator, the expression 5 ​ ____________________
  
  ​
may be written as a large algebraic fraction. a(a 2 1)
2​a​2​
​5 ______
​ ​
Simplify the similar terms in the numerator position. a(a 2 1)
2​a​​​ ​2​​
Simplify the resulting expression, if possible. 5 _______
​​   ​​
​a​(a 2 1)
Specify the restriction(s) together with the 2a
equivalent simplified expression. ​5 ​_____,​​ where a  0 and a  1
(a 2 1)

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NOTE: Even if the simplified expression has only one restriction, all those named at the start must be applied.

CAUTION!
When subtracting an algebraic expression that has a polynomial
as the numerator, each of the monomials must be subtracted.
Example:

3
___ 3x 1 5​x​3​2 5 ___3 23x 2 5​x​3​2 (25) 25​x​3​2 3x 1 8
​ 2 ​2 _________
​ ​5 ​ 2 ​1 ​_____________
   ​5 __________
​ ​
2​x​ ​ 2​x​ ​
2
2​x​ ​ 2​x​ ​
2
2​x​2​

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Algebraic Identity
An identity is an equation that is always true regardless of the value attributed to the variables.

Example:
The result of the previous example allows the following identity to be written.
3
____ 2a 2 1 1 2a
​ ​ ​1 ​_____​2 ______
​ ​5 ____
​ ,​​ where a  0 and a  1
a21 a a(a 2 1) a 2 1
Below is a validation of this identity when a 5 2.
3
_____ 2(2) 2 1 1 2(2)
​ ​1 ​______​2 ________
​ ​5 _____
​ ​
(2) 2 1 (2) (2)((2) 2 1) (2) 2 1
3 3 __ 1 4
​​ __ ​​ 1 __ ​ ​2 ​ ​5 __
​​
1 2 2 1
454

Situation 2.1
PRACTISE

Acquisition A
6 Calculate the sum or the difference of the following rational expressions. Clearly show each step
in your work.
2
____ x12 8 5 30 2 4y
a) ​​ ​1 ​____​1 ______
​ ​​ b) ​​__​2 ​______​
x14 x x(x 1 4) y 2y 1 5

7 Validate the expressions calculated above as follows.

a) By replacing x with 2
b) By replacing y with 2
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3. Factoring a Perfect Square Trinomial

You have obtained identities by using addition and subtraction of rational expressions. To solve the
situational problem of the prodigy, you also need to factor certain special trinomials. Various techniques can
be used in the factorization of polynomials. The form of the polynomial often provides clues about which
technique is best to use. Below is a special trinomial derived from the square of a binomial; this is called a
perfect square trinomial.

REFRESHER EXERCISES
REMINDER PAGE 119, QUESTIONS 18 AND
19
The Distributive Property and Product of Two Binomials
To determine the product of two binomials, simply multiply each term of the first binomial by each
term of the second binomial.

Example:
To multiply 2x 1 3 by 4x 2 2, proceed as follows:

(2x 1 3)(4x 2 2) 5 (2x 3 4x) 1 (2x 3 22) 1 (3 3 4x) 1 (3 3 22) 5 8x2 2 4x 1 12x 2 6
This book is the property of Dickson Joseph.

Since there are four multiplications to perform, the result is a polynomial with four terms.
In this case, the polynomial can be returned to a trinomial by simplifying the two x terms.

8x2 2 4x 1 12x 2 6 5 8x2 1 8x 2 6

8 The illustration to the right represents a square whose sides measure

a 1 b units. Determine two equivalent algebraic expressions that
a b
represent the area of the square.
a

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9 Calculate the following expressions.
a) (x 1 8)2 b) (x 2 2y)2

c) (2xy 2 3)2 d) (x2 1 2x)2

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10 Consider the trinomial 4x2 1 20x 1 25.

a) What is the square root of 4x2?
b) What is the square root of 25?
c) How can you evaluate the term 20x based on the results of a) and b)?

REMEMBER

Perfect Square Trinomials

A perfect square trinomial always comes from the square of a binomial. You can recognise this because
it is made up of three terms with this characteristic.
(a 1 b)2 5 a2 1 2ab 1 b2

Square of the Double the product of Square of the

binomial’s 1st term the binomial’s 2 terms binomial’s 2nd term

Situation 2.1
Acquisition A
To validate that a trinomial is in fact a perfect square trinomial, follow the steps below.
• Calculate the square root of the first term.
Example:
• Calculate the square root of the third term.
(4x 1 1)2 5 16x2 1 8x 1 1
• Calculate the double of the product of these
two square roots. The square This term equals The square
If the product of this calculation is equivalent to root of the the double of the root of the
the middle term, it is a perfect square trinomial. 1st term product of 4x 3rd term is 1.
is 4x. and 1.

NOTE: This procedure works when the trinomial is written in the order presented above.

Factoring a Perfect Square Trinomial

It is possible to factor a perfect square trinomial by finding the square of the binomial that it is equal to.
© SOFAD / All Rights Reserved.

Example:
Follow the steps below to factor the perfect square trinomial k2 1 4kv 1 4v2:
• The square root of the first term is k.
• The square root of the third term is 2v.
• The double of the product of the two roots is 2 3 k 3 2v 5 4kv
(which matches the middle term of the perfect square trinomial).
• Thus, k2 1 4kv 1 4v2 5 (k 1 2v)2.
Therefore, the factors of the perfect square trinomial are k 1 2v and k 1 2v.

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PRACTISE

11 Circle the trinomials below that are perfect squares. CAUTION!

Then express each trinomial you circled as a squared
binomial. The square root of a number is
always positive. For example, the
a) x2 2 6x 1 9 b) x2 1 2x 1 4 square root of 49 is 7. However,
the equation x2 5 49 has two
solutions: x 5 7 and x 5 27. These
numbers are called the roots of
the equation. Therefore, when you
c) 4x2 112x 1 9 d) 36x2 2 84x 1 49
want to determine the roots for a
perfect square trinomial, you must
also provide for negative values
since one of the terms of the
square of the binomial may have
12 What term must be added to each expression to make a a negative sign.
perfect square trinomial?
a) x2 1 10x 1 b) 4x2 1 1 9y2 c) 2 30x 1 9
This book is the property of Dickson Joseph.

4. Simplifying a Rational Expression Using a Perfect Square

Trinomial
Here is the final piece of information you need to expose the prodigy. All that remains is to combine the
factorization of perfect square trinomials in order to simplify the algebraic fractions.

Consider the following algebraic expression: ___________

64​x​2​1 16x 1 1
13 ​​  8x 1 1 .​​
a) Is the numerator a perfect square trinomial? If so, factor it.

b) Can you simplify the algebraic expression?

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REMEMBER

Simplifying Rational Expressions

To simplify an algebraic fraction, the numerator and denominator must be made up of the products
of factors.
A special identity may be used to simplify algebraic expressions.

Example:
x19 x
The steps below show how to use a perfect square trinomial to simplify ​_________​1 ____
​ .​​
(x 1 5)(x 1 3) x 1 3

Explanations Example
Begin the same way as you would to add rational expressions.
x19 x
The common denominator is (x 1 3)(x 1 5). 5 ​_________​1 ____
​ ​
(x 1 5)(x 1 3) x 1 3
Therefore, the restrictions are x  23 and x  25.
Perform the necessary multiplications to get the equivalent x19 x(x 1 5)
5 ​_________​1 _________

Situation 2.1
​ ​

Acquisition A
algebraic fractions. (x 1 5)(x 1 3) (x 1 3)(x 1 5)
x19 ​x​2​1 5x
Simplify all the polynomials in the numerator. 5 ​_________​1_________
​ ​
(x 1 3)(x 1 5) (x 1 5)(x 1 3)
Place the entire algebraic expression in the same fraction. x 1 9 1 ​x​2​1 5x
5 ​____________
   ​
Be careful to leave the denominator in factor form. (x 1 5)(x 1 3)
​x​2​1 6x 1 9
Then, add the similar terms in the numerator position. ​5 _________
​ ​
(x 1 5)(x 1 3)
If possible, factor the numerator. For example, here ​(x 1 3)​​2​
5 ​_________​
x2 1 6x 1 9 5 (x 1 3)2. (x 1 5)(x 1 3)
(x 1 3)​2​
Simplify the factored expression by identifying common factors. 5 ____________
  
​​   ​​
(x 1 5)​(x 1 3)​
(x 1 3)
Apply the restrictions in the final simplified expression. ​5 _____
​ ,​​ where x  23 and x  25
(x 1 5)

PRACTISE
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14 Simplify the following expressions.

2x 6 6x 3​x​2​2 1
​______​1 _____
a) ​ ​ ​​ b) ​​_____​2 ____________
​  ​
x 1 1.5 2x 1 3 3x 2 1 ​(2x 2 1)(3x 2 1)​

You now have all the information you need to reveal the prodigy’s trick and expose this type of performer
claiming to have paranormal powers.

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SOLUTION
You are now able to solve Situational Problem 2.1. LAWEXPRESSIONS
RATIONAL OF COSINES
TRIGONOMETRIC
PERFECTFORMULAE FOR AREA
SQUARE TRINOMIALS
SITUATION 2.1 HERON'S OF
DIFFERENCE FORMULA*
SQUARES

The Prodigy SP 2.1

During a variety show, a performer

You must demonstrate that the performer does appears on stage and introduces himself
as a mental calculation prodigy. He claims
to be able to solve complex mathematical
TASK

not really have a prodigious talent and identify problems in a fraction of a second without
the help of a calculator or even a pencil.
He challenges anyone in the room to solve

the mathematical procedure he uses to solve a complex mathematical expression faster

than him after replacing the variable with
any number.

the equation so quickly. The prodigy asks an audience member to pick a number, which he will substitute
into the expression on the screen. She picks number 7. The prodigy answers: _7 .
9

The prodigy then invites other audience members to give him different
numbers. Someone shouts out “Five!” Someone else says “Three!” In next to no
7 5
time, the prodigy answers: __, __.
5 3

TASK
You must demonstrate that the performer does not really have a prodigious talent
and identify the mathematical procedure he uses to solve the equation so quickly.

© SOFAD / All Rights Reserved.

This book is the property of Dickson Joseph.

66 CHAPTER 2 – Algebraic Expressions

SITUATIONAL PROBLEM FROM PAGE 66

Summary of the Facts
• The rational expression given by the prodigy is as follows:
n11
_____ 1 2
​​   ​​ 1 _____
​​   ​​ 1 ________
​​   ​​ 5 ?
n n 1 2 n(n 1 2)
9
• If the chosen number is 7, the answer is __
​ .​​
7
7
• If the chosen number is 5, the answer is __
​ .​​
5
5
• If the chosen number is 3, the answer is __
​ .​​

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3

STRATEGY Formulating a conjecture

Before attempting to solve the problem, it may be useful to formulate a conjecture to help guide your thought
process. For example, by examining the algebraic expression and the answers given by the prodigy, you can get an
idea of the simplified expression you will end up with. This conjecture can be used to develop your procedure.

Solution

78 CHAPTER 2 – Algebraic Expressions

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Solution (cont’d)

Situation 2.1
Solution
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Procedure:

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ACQUISITION B Mathematical knowledge

targeted:
•• recognizing a difference
of two squares
1. Difference of Two Squares •• factoring using the
difference of two squares
In Acquisition A, you calculated the square of a binomial and learned how
to recognize a perfect square trinomial. Other special identities also exist.
In Acquisition B, you will learn what happens when the sum of two terms is
multiplied by the difference of these same two terms.

1 a) Expand the following algebraic expression: (a 1 b)(a 2 b).

b) What do you notice?

REMEMBER
This book is the property of Dickson Joseph.

Difference of Two Squares

The product of the sum of two terms and the difference of these same terms is equal to the difference
of the two squares of these terms.
(a 1 b)(a 2 b) 5 a2 2 b2

Sum of Difference of the Difference of the two

2 terms same terms squares of these terms

The resulting equation is an identity; regardless of the values given to the variables, equality is maintained.

Example:
(3x2 1 2y)(3x2 2 2y) 5 9x4 2 6x2y 1 6x2y 2 4y2

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5 9x4 2 4y2
Sum of 3x2 Difference Difference of the
and 2y of 3x2 and 2y squares of 3x2 and 2y

PRACTISE

2 Perform the following operations.

a) (x 1 3)(x 2 3) 5
b) (4x 1 y)(4x 2 y) 5

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2. Factoring Using the Difference of Two Squares

Now that you are familiar with this new identity, it is often useful to recognize it quickly—either to determine
the restrictions of a rational expression or to simplify it. The following questions will help make this clear.
​x​2​ 2 9
3 Consider the following rational expression: ​_________ .​​
3​x​2​ 2 9x

a) Determine its restrictions.

b) Simplify it.

REMINDER REFRESHER EXERCISES

PAGE 119, QUESTION 20
Simple Factorization

Situation 2.1
Acquisition B
Breaking a polynomial down into factors means writing it in the form of a product. This can be done
using simple factorization.

Example:
Factor the following expression: 12a3x2 1 15a2x 2 9a.
Since all the terms are divisible by 3a, we can perform the following simple factorization.
12a3x2 1 15a2x 2 9a 5 3a(4a2x2 1 5ax 2 3)

REMEMBER

Factoring Using the Difference of Two Squares

A binomial that is the difference of two squares can be factored into the product of two binomials.
© SOFAD / All Rights Reserved.

a2 2 b2 5 (a 2 b)(a 1 b)

Example:
9x2 2 16 5 (3x 1 4)(3x 2 4).

The square root The square Sum of the Difference

of the 1st term root of the square of the
is 3x. 2nd term is 4. roots square roots

ANSWER KEY PAGE 163 81

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PRACTISE

4 Factor the following differences of two squares. Tip

a) 16x2 2 1 b) 36 2 4a2 c) 225 1 x2 Remember: To check
your answer, replace the
variable by any number
other than 0 in the
initial expression and in
the final expression. If
the results are identical,
the expressions are
5 If the area of a rectangle is represented by the expression 64x2 2 25,
equivalent!
which algebraic expressions may be used to represent the length of
each side?

6 Simplify the rational expressions below. Remember to name the restrictions.

3x 1 1
_______ 1.44 2 ​a​2​
a) ​
​ 2 ​​ b) ​​__________​
9​x​ ​ 2 1 a 1 1.2
This book is the property of Dickson Joseph.

DID YOU KNOW?

Indian mathematicians were the first to use a specific symbol (zero: 0)
to denote an absence or void. Until then, 0 was not considered a
number. The smallest possible number was 1. Some historians believe
that the invention of 0 was the key to the enrichment of arithmetic and
algebraic calculations.

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82 CHAPTER 2 – Algebraic Expressions

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CONSOLIDATION
1 Using special identities, do the following calculations in your head, then show your work.
a) 522 5
b) 392 5 Tip
c) 2432 2 2422 5 Algebraic identities can even be used in cases where
there is no variable. They can be used to make mental
d) 61 3 59 5
calculations.
e) 63 3 57 5
Example:
It is easy to calculate 412 or 24 3 16 as follows:
412 5 (40 1 1)2 5 402 1 2 3 40 3 1 1 12 5
1600 1 80 1 1 5 1681
24 3 16 5 (20 1 4)(20 2 4) 5 202 2 42 5

Situation 2.1
consolidation
400 2 16 5 384

3 2x 2 1 ______1 2x
2 In the expression ____
​ ​1 _____
​ ​2 ​ ​5 ____
​ ​, where x  0 and x  1, verify the equation by
x21 x x(x 2 1) x 2 1
replacing the variable with the following values:
1
a) x 5 21 b) x 5 __
​​
2

3 Name the restriction(s) that must be given to the variable x so that the following rational expressions are
properly defined.
x x22
​______​
a) ​ ​_______​
b) ​
x11 2x 2 1
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x11 3​x​2​
​_________​
c) ​ ​_________2 ​
d) ​
3x(x 2 2) ​(2x 2 6)​​ ​
4 Simplify the following rational expressions.
1
__ a 3x 1 3
​ ​1 ___
a) ​ ​ ​ b) ​​ ______  ​​
a 2 ​a​2​ ​x​​  2​ 2 1

​x​​  2​1 4x 1 4 12x x 2 5 _______

13x 2 4
c) ​​ __________  ​​ d) ​​_____​2 ____
​ ​1 ​ ​
​x​​  2​2 4 3x 1 4 x x(3x 1 4)

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5 Perform the following operations.

a) (x 1 8)2 5 b) (5x 2 1)2 5
c) (x 1 5)(x 2 5) 5 d) (x 2 2y)2 5
e) (2xy 2 3)2 5 f ) (2x 1 y)(2x 2 y) 5
g) (x2 1 2x)2 5 h) (3 2 5x2)(3 1 5x2) 5

6 Circle the trinomials below that are perfect squares. Then express each trinomial you circled as a
squared binomial.

x2 2 14x 1 49 x2 1 4x 1 16 4x2 1 20x 1 25

36x2 2 6x 1 0.25 4x2 2 24x 1 9 9x2 1 12x 1 4

This book is the property of Dickson Joseph.

7 For each expression below, what term must be added to make a perfect square trinomial?
a) x2 1 18x 1 b) 9x2 1 1 100y2 c) 2 50x 1 25

8 Factor the following binomials.

a) 9x2 2 16 5 b) x2 2 4y2 5

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c) x2y2 2 25 5 d) x4 2 1 5

84 CHAPTER 2 – Algebraic Expressions

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9 Simplify the rational expressions after factoring their numerator and/or denominator. In each case,
specify the restrictions you need to apply to the variable x so that the simplified expression and the
initial expression are equivalent.
​x​2​ 1 2x 6x 1 9
a) ​​________ ​​ b) ​​_________ ​
2​x​2​ 4​x​2​ 1 6x

​x​2​ 2 4 ​x​2​ 2 2x 1 1
​________
c) ​ ​​ d) ​​____________ ​
​x​2​ 1 2x ​x​2​ 2 1

Situation 2.1
consolidation
4​x​2​ 1 2x
​_____________
e) ​ ​
4​x​ ​ 1 4x 1 1
2

10 Perform the following operations.

1
_______ x11 1 x11
a) ​
​2 ​​1 ______
​ ​​ b) ​​_______ ​​2 ______
​ ​
​x​ ​ 2 1 x 2 1 ​x​ ​ 2 1 x 2 1
2
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c) ​ (
​2
1
​ _______ )(
​x​ ​ 2 1 x 2 1
x11
​ ​​​ ______
​ )
​ ​​ ( 1
d) ​​ _______
​2
​x​ ​ 2 1) ( x11
​ ​​4 ​ ______
​
x21 )
​​

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LAW OF COSINES
MULTIPLYING
TRIGONOMETRIC TWO POLYNOMIALS
FORMULAE FOR AREA
SITUATION 2.2 HERON'S
DIVIDING A POLYNOMIAL BY FORMULA*
A BINOMIAL

The Telepath SP 2.2

Soon after the prodigy’s act, another

performer comes on stage, claiming to be
telepathic. Telepathy means sending thoughts
to another person using only one’s mind—
without words or gestures. Most scientists
do not believe in telepathy, or are highly
skeptical, because there is no reliable
evidence of its existence.
This book is the property of Dickson Joseph.

To demonstrate her gift, the telepath asks for five volunteers

to play the following game:
“I’m going to send you a secret number with my mind. Once you
have visualized this secret number in your mind, follow these
steps. We will then find out whether each of you has seen the
correct number.”
1. Calculate the cube of the secret number.
2. To the result, add the triple of the square of the secret number.
3. To that result, add three more than the secret number.
4. Multiply the answer by two less than the secret number.

© SOFAD / All Rights Reserved.

5. Divide the answer by one more than the square of the secret number.
6. Subtract the answer from the sum of the secret number and its square.
7. Write the answer on the blank card without showing it to anyone.
You suspect that this result has nothing to do with telepathy and decide to reveal the performer’s trickery.
TASK

You must demonstrate, mathematically, that this performance has nothing to do with
telepathy and that the result is due to operations performed on algebraic expressions.

86 CHAPTER 2 – Algebraic Expressions

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EXPLORATION
The questions in this exploration activity will help you establish a series of operations to be applied to
algebraic expressions. To solve these expressions, you should review the order of operations.

1 a) To determine the numbers used in the telepath’s procedure, it is useful to create a schematic diagram
based on a specific number. Here, the value 4 has been chosen. Complete the diagram.
Cube Triple of the square

Number
Three more than Two less than
the number 4
the number

Situation 2.2
exploration
One more than the Sum of the number
square of the number and the square

STRATEGY Schematization

Using a schematic diagram can help you understand the components of a complex procedure and determine
the series of operations associated with a situation. Never hesitate to draw a diagram to help you solve a difficult
situational problem.

b) Use the telepath’s instructions to determine the arithmetic expression that corresponds to the series
of operations carried out by the volunteers if the secret number is 4.
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2 Determine the value of the expression obtained in question 1.

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REMINDER REFRESHER EXERCISES

PAGE 120, QUESTIONS 21 AND 22
Order of Arithmetic Operations
To determine the value of an expression containing several operations, you must proceed in a clearly
defined order.

Example:
What is the value of 32 2 4(9 ÷ 3) 1 32?
Tip
BEDMAS
Order of Operations Example
1. Perform the operations in brackets. 5 32 2 4 3 3 1 32 Use this acronym to
help you remember the
2. Calculate the exponents. 5 32 2 4 3 3 1 9
order of operations.
3. Perform the divisions and multiplications. 5 32 2 12 1 9
B: Brackets
4. Perform the additions and subtractions. 5 29
E: Exponents
D: Division
M: Multiplication
A: Addition
S: Subtraction
3 a) Perform the same calculations again, this time with 2 as
This book is the property of Dickson Joseph.

the secret number..

b) What conclusion can you draw from the resulting number?

4 Use a letter of your choice to write an algebraic expression that corresponds to the telepath’s
instructions. Save this expression for your solution.

In the series of operations that will be used to arrive at the final number, the denominator of the algebraic © SOFAD / All Rights Reserved.

fraction must be a binomial—a type of expression we have not covered yet. In the acquisition activity that
follows, you will divide a polynomial by a binomial, allowing you to expose another of these performers.

88 CHAPTER 2 – Algebraic Expressions

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ACQUISITION A Mathematical knowledge

targeted:
•• multiplying polynomials
•• dividing a polynomial by
1. Multiplying Polynomials a binomial without a
remainder
Before you learn about dividing a polynomial by a binomial, you must review the
concept of multiplying polynomials. The questions below will help with this review.

1 Simplify the following algebraic expressions.

a) 6x2y5 3 (22x3y) b) 7x(2x 2 3) c) (5x2 1 3)(x 2 4)

2 a) Multiply each term of the binomial by each term of the trinomial, then simplify the result.

Situation 2.2
(x2 2 4x)(x2 1 2x 17)

Acquisition A
b) What is the degree of your polynomial?

REMINDER REFRESHER EXERCISES

PAGE 120, QUESTIONS 23 AND 24
Terms and Degrees
A term refers to each of the elements that make up a polynomial. This may be a constant or an
algebraic expression consisting of a coefficient and one or more variables that are multiplied.
The degree of a monomial is the sum of the exponents of the constituent variables. The degree of a
polynomial corresponds to the highest degree of its terms.

Example:
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• 2xy is a monomial resulting from the multiplication of the number 2 and its variables x and y.
• The degree of 4a2b is 3.
• In the polynomial 3a2b5 1 2ab3 2 1, the degree is 7 and there are three terms.

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REMEMBER

Multiplying Two Polynomials

To multiply two polynomials, simply apply the distributive property by multiplying each term of the
second polynomial by each term of the first.

Example:
The following example is one way to multiply two second-degree trinomials.

(x2 2 4x 1 1)(x2 1 2x 1 7) 5 x4 1 2x3 1 7x2 2 4x3 2 8x2 2 28x 1 x2 1 2x 1 7

After simplifying the similar terms, a fourth-degree polynomial is obtained: x4 2 2x3 2 26x 1 7.

PRACTISE
Tip
3 Perform the following multiplications. The order of the terms in a polynomial has no effect
This book is the property of Dickson Joseph.

a) (9x 1 3)(2x 2 5x 1 2)
5 3 on its value.

b) (3a2 2 2a 2 5)(a3 2 a 1 2) Example:

3x 1 4 1 x2 5 x2 1 3x 1 4
c) (0.5y3 1 0.25)(y4 2 3y2 1 6y 1 4)
However, according to mathematical convention, it is
preferable to write a polynomial with its terms in order
from the term with the largest degree to the term with
the smallest. This order is ideal for dividing polynomials.

Furthermore, if there are several variables in the same

term, it is best to write these in alphabetical order.
Therefore, even though 3xy2 is equivalent to 3y2x, it is
better to write this term as 3xy2. This will make it easier
to identify similar terms.

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90 CHAPTER 2 – Algebraic Expressions

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2. Dividing Polynomials
You have already learned how to divide a polynomial by a monomial. Let us now consider a more complex
operation: dividing a polynomial by a binomial. This is what you will study in the next few pages.

4 Simplify the following fractions.

4​x​3​
___ 3e
___ x​y​4​
a) ​
​ ​​5 b) ​
​ 4 ​​5 ​____
c) ​ ​5
2x 6​e​ ​ ​x​2​y
5 Perform the following divisions.
a) (4x2 2 2x) 4 4x 5 b) (8a2 2 4a 1 1) 4 2 5
c) (27x4 1 6x3 2 3x2 2 3x) 4 22x 5

REMINDER REFRESHER EXERCISES

PAGE 120, QUESTION 25

To divide a polynomial by a monomial, simply

divide the coefficients by each other, and divide Example:

Situation 2.2
Acquisition A
the variables by subtracting their respective 6​a​3​ 3​a​2​ ___
a
(6a3 2 3a2 1 a) 4 2a 5 ___
​ ​2 ___
​ ​1 ​ ​
exponents. The terms may be expressed as 2a 2a 2a
fractions, then the properties of algebraic 3 1
5 3a2 2 __
​ ​a 1 __
​​
simplification can be applied. 2 2

6 Without using a calculator and based on the principle of long division, perform the following divisions.
a) 1250 5 b) 6936 12

7 a) Taking inspiration from the calculations carried out in question 6, can you perform the following
© SOFAD / All Rights Reserved.

division? 2x2 2 7x 2 4 x 2 4

b) How would you proceed?

STRATEGY Use different types of representations to interpret the facts of a problem

To learn, you need to try things out. If you are missing some of the tools required to solve a problem, drawing
on your existing knowledge and your intuition may be enough to help you solve it. In this way, you will acquire
new knowledge.

ANSWER KEY PAGE 166 91

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REMINDER REFRESHER EXERCISES

PAGE 120, QUESTION 26
Division
Below is a description of the algorithm for division in four steps.
1) Perform the division on the first two digits of the
dividend (i.e. 25) only. To do this, determine the Example:
number of times the divisor goes into this number 2576 12 Divisor
Dividend
(12 goes into 25 twice). – 24 214 Quotient
17
2) Multiply the divisor by the number obtained by the – 12
previous operation (2 3 12). 56
– 48
3) Subtract the result from the initial number (25 2 24). 8 Remainder
4) Bring down the next digit (7), then begin the process again.

REMEMBER

Dividing a Polynomial by a Binomial

This book is the property of Dickson Joseph.

To divide a polynomial by a binomial, certain steps need to be repeated:

• Identify a term that, when multiplied by the first term of the binomial divisor, gives a product equal
to the first term of the polynomial to be divided. This is the first term of the quotient.
• Multiply the term you have identified by the binomial divisor.
• Write the resulting binomial below the polynomial.
• Subtract this binomial from the polynomial to obtain a new polynomial.
Repeat until there are no more terms in the dividend.

NOTE: The initial polynomial must be ordered correctly.

Example:

© SOFAD / All Rights Reserved.

The following algorithm can be used to perform this division: (2x2 1 7x 2 15) ÷ (x 1 5).
Dividend Divisor

2x2 1 7x 2 15 x15
2 (2x2 1 10x) 2x 2 3
2 3x 2 15
2 (23x 2 15)
Quotient
0
Thus, you get (2x2 1 7x 2 15) 4 (x 1 5) 5 2x 2 3.

NOTE: When the remainder is 0, we say that the polynomial is divisible by the binomial. In this case, the
polynomial can be factored and the binomial is one of its factors.
2x2 1 7x 2 15 5 (x 1 5)(2x 2 3).

92 CHAPTER 2 – Algebraic Expressions

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PRACTISE

8 Showing your work, perform the following divisions.

a) (2x3 1 5x2 1 7x 1 10) 4 (x 1 2) b) (2x2 2 7x 2 4) 4 (x 2 4)

9 Simplify the following rational expression. CAUTION!

4a4 1 4a3 2 a 2 1
_______________
​​     ​​
4a3 2 1 Polynomials included in a division may have
negative coefficients. This does not change
the steps in the calculation. However, ensure
that you pay attention to the rules that apply

Situation 2.2
Acquisition A
to signs associated with the various
intermediate operations to be carried out
(division, multiplication, subtraction).

It is now time to solve Situational Problem 2.2 in this chapter. You should be able to show that there is
nothing extraordinary about these so-called telepathic powers.
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SOLUTION
You are now able to solve Situational Problem 2.2. LAW OF COSINES
MULTIPLYING
TRIGONOMETRIC TWO POLYNOMIALS
FORMULAE FOR AREA
SITUATION 2.2 HERON'S
DIVIDING A POLYNOMIAL BY FORMULA*
A BINOMIAL

The Telepath SP 2.2

Soon after the prodigy’s act, another

performer comes on stage, claiming to be
telepathic. Telepathy means sending thoughts
to another person using only one’s mind—
without words or gestures. Most scientists

You must demonstrate, mathematically, that this do not believe in telepathy, or are highly
skeptical, because there is no reliable
evidence of its existence.
TASK

performance has nothing to do with telepathy To demonstrate her gift, the telepath asks for five volunteers

and that the result is due to operations carried

to play the following game:
“I’m going to send you a secret number with my mind. Once you
have visualized this secret number in your mind, follow these

out on algebraic expressions. steps. We will then find out whether each of you has seen the
correct number.”
1. Calculate the cube of the secret number.
2. To the result, add the triple of the square of the secret number.
3. To that result, add three more than the secret number.
4. Multiply the answer by two less than the secret number.

© SOFAD / All Rights Reserved.

5. Divide the answer by one more than the square of the secret number.
6. Subtract the answer from the sum of the secret number and its square.
7. Write the answer on the blank card without showing it to anyone.
You suspect that this result has nothing to do with telepathy and decide to reveal the performer’s trickery.

TASK
You must demonstrate, mathematically, that this performance has nothing to do with
telepathy and that the result is due to operations performed on algebraic expressions.
This book is the property of Dickson Joseph.

86 CHAPTER 2 – Algebraic Expressions

SITUATIONAL PROBLEM FROM PAGE 86

Summary of the Facts

• The result always seems to be 6.
• According to the exploration activity, the algebraic expression that corresponds to the series of operations
to be carried out may be written as follows:
(​x​3​1 3​x​2​1 (x 1 3))(x 2 2)
​(​x​2​1 x) 2 ___________________
​ ​
​x​2​1 1

Solution

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94 CHAPTER 2 – Algebraic Expressions

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Solution (cont’d)

Situation 2.2
Solution
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Conclusion:

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ACQUISITION B Mathematical knowledge

targeted:
•• dividing a polynomial by a
binomial with a remainder
1. Dividing a Polynomial by a Binomial
With a Remainder
To conclude this chapter, we will explore the division of a polynomial by a binomial with a remainder.
Just like numerical fractions, dividing algebraic expressions does not always produce an integer.

1 Perform the following divisions.

a) 127 7 b) 25,348 15

2 a) Perform the following algebraic division.

(2x3 1 5x2 1 7x 1 6) 4 (x 1 2)
This book is the property of Dickson Joseph.

b) With reference to the algorithm of arithmetic division, how can the remainder be written in the
quotient (answer) in the case of algebraic division?

DID YOU KNOW?

© SOFAD / All Rights Reserved.

The symbol used in long division originally comes from what is called Euclidean
division.

Euclid, a Greek mathematician and philosopher who lived in the 3rd century BC,
founded the most celebrated school of Antiquity. His works primarily consist of a
13-volume encyclopedia on topics including geometry and arithmetic.

Dividend 119 11 Divisor

2 11 10 Quotient
09
2 00
Remainder 9

96 CHAPTER 2 – Algebraic Expressions

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REMEMBER

The Remainder of a Polynomial Divided by a Binomial

When applying division to natural numbers, the remainder is always a positive number that is less than
the divisor. The convention regarding remainders is different for polynomials. When dividing polynomials,
you must continue the process until the degree of the remainder is less than the degree of the divisor.

Example:
Consider the division of x3 2 3x2 1 6 by x 2 2.
CAUTION!
x3 2 3x2 1 0x 1 6 x22
Terms may be missing from the process.
2 (x3 2 2x2) x2 2 x 2 2
2x2 1 0x Example:
2 (2x2 1 2x) When you divide x3 2 3x2 1 6 by x 2 2,
22x 1 6 you can see that the dividend does not
2 ( 2x 1 4)
2 have any x terms.
2 In this case, to aid calculation, it is a good

Situation 2.2
Acquisition B
idea to leave a blank space or, even better,
The remainder of this division is 2.
to add a term 0x between the x2 term and
the constant term, as shown in the example
on the left. This way, a column is set aside
for the x terms, which will appear during
the calculation.
PRACTISE

3 Perform the following divisions.

a) (8a3 1 22a2 2 21a 2 3) 4 (2a 1 7) b) (3x3 2 6x2 1 10) 4 (x 2 1)
© SOFAD / All Rights Reserved.

2. Checking the Division of Polynomials

It is always a good idea to check that the resulting quotient is correct. The following section provides ways of
carrying out these checks.

4 In the previous acquisition activity, the quotient 2x 1 1 resulted from the division of 2x2 2 7x 2 4 by x 2 4.
How can you check that this quotient is correct? Use algebraic manipulation to back up your answer.

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5 In question 2 of the acquisition activity, the quotient 2x2 1 x 1 5, with remainder 24, was the result of
dividing 2x3 1 5x2 1 7x 1 6 by x 1 2.
How can you check that this quotient is correct? Justify your answer using algebraic manipulation.

REMEMBER

Writing Algebraic Equalities when Dividing Polynomials

An algebraic quotient resulting from the division of a polynomial by a binomial may be written in
different forms. It can be expressed as the answer to the division or as a factor of the dividend.

Division Without Remainder

As a quotient: As factors:
(2x2 2 7x 2 4) 4 (x 2 4) 5 (2x 1 1) (2x2 2 7x 2 4) 5 (x 2 4) (2x 1 1)
This book is the property of Dickson Joseph.

Dividend Divisor Quotient Dividend Factor Factor

Division With Remainder

However, in the case of divisions in which the algorithm produces a remainder, the two forms of
expressing the answer, either as a division or as a multiplication, are not so simple.
As a quotient:
24
(2x3 1 5x2 1 7x 1 6) ÷ (x 1 2) 5 (2x2 1 x 1 5) 1 ______
​ ​
x12

Remainder
_______
Dividend Divisor Quotient ​ ​
Divisor

© SOFAD / All Rights Reserved.

As an equivalent algebraic expression:
(2x3 1 5x2 1 7x 1 6) 5 (x 1 2)(2x2 1 x 1 5) 1 (24)

Dividend Divisor Quotient Remainder

98 CHAPTER 2 – Algebraic Expressions

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PRACTISE

6 Use two different ways to check the result of question 3 above.

a) By multiplying the resulting factors. b) By replacing the variable x with a
numerical value.

STRATEGY Checking algebraic operations

Situation 2.2
Acquisition B
There are many strategies you can use to validate algebraic identities. Earlier, you learned to replace the variable of
the identity with a numerical value. Sometimes it can be useful to perform the reverse operation. For example, if a
polynomial has been divided by a binomial, multiplying the divisor by the quotient should produce the dividend. In
the case of a division with a remainder, the remainder must be added to the multiplication to obtain the dividend.
© SOFAD / All Rights Reserved.

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CONSOLIDATION
1 Perform the following multiplications.
a) (x 2 3)(x2 1 3x 1 2) b) (2x 1 1)(x2 2 3x 1 3)

c) (3x 2 2)(4x2 2 x 2 3) d) (2x 1 3)(4x2 2 6x 1 9)

2 Perform the following divisions. Then write the result using an equation.
a) 4x2 – 8x – 6 2x + 1 b) 9x2 – 15x + 4 3x – 4

3 Calculate the quotient for the following divisions.

This book is the property of Dickson Joseph.

a) (2x3 2 x2 2 9x 1 11) 4 (2x 2 3) 5 b) (x3 2 2x2 1 4x 2 8) 4 (x 2 2) 5

c) (x3 1 6x 1 100) 4 (x 1 4) 5 d) (x3 2 3x2 1 5) 4 (x 1 1) 5

© SOFAD / All Rights Reserved.

4 a) Which of the following polynomials is divisible by x 1 3? Justify your answer.
(A) x2 1 9 (B) x2 2 9x 2 27 (C) 2x2 2 3x 2 27

b) Factorise the polynomial you identified in a).

100 CHAPTER 2 – Algebraic Expressions

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5 The dimensions, in decimetres, of the two boxes below are shown using algebraic expressions,
where x represents the height of Box A.

(A) (B)

x
x1 x3
x4

x1 x3

Which box has the greater capacity? Justify your answer.

Situation 2.2
consolidation
6 Victor says that the polynomial x3 2 8 is necessarily divisible by x 2 2, because x3 4 x gives x2 and
28 divided by 22 is 4. He therefore concludes that the quotient is x2 1 4. His sister, Judith, tells him

that he is both right and wrong about this. Explain Judith’s point of view.

7 A famous Chinese book entitled Jiuzhang Suanshu, known in English as The Nine Chapters on the
Mathematical Art, contains problems that have been studied for centuries. In the year 263 AD,
Chinese mathematician Liu Hui published solutions to some of them. The figures below illustrate
one of these problems.

b
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Using the information provided, demonstrate that the shapes have equivalent areas—that is, the sum of
the area of the first two squares is equal to the area of the last: a2 1 b2 5 c2.

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8 Hamid, a magician’s apprentice and math enthusiast,

wants to impress his master with an algebra-based trick.
He formulates the conjecture that, if x is a positive number
greater than 1, regardless of the binomial by which the
expression 324x3 2 45x2 2 99x 2 10 is divided, the result
is always a number greater than x. As examples, he cites
the binomials 3x 2 5 and 9x 1 1.
After checking that dividing by the first binomial produces
a quotient greater than x and that the same is true for the
second binomial, check that Hamid’s conjecture is true for
any binomial.
This book is the property of Dickson Joseph.

9 Magali described the dimensions of a rectangular band

using two algebraic expressions. She then split the band
into two pieces so that the relationship between the areas
of Piece A and Piece B is equal to __________
3​x​2​ 2 2x 2 8
​​ 3​x​2​ 2 10x 1 8 ​​.
The illustration below shows the band without its dimensions.

A B

a) Determine the value of the rational expression if:

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1) x 5 0;
2) x 5 1.
b) Does something unusual happen if x 5 2?

102 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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c) Referring to your answer to b), which binomial are the numerator and the denominator both divisible
by? Explain your answer.

Simplifying the rational expression would make it easier to calculate the relationship between the areas
of the two pieces.
d) Referring to your answer to c), factor the numerator and the denominator of this rational expression.
3​x​2​ 2 2x 2 8
______________
​  
   ​​5
3​x​2​ 2 10x 1 8
e) What restrictions must be applied to the variable x?

Situation 2.2
consolidation
f ) Express the relationship between the areas using a simplified fraction, then determine the value of
the relationship if x 5 5.

g) Taking all the above into account, can you deduce the algebraic expressions that Magali used to
describe the dimensions of the band at the outset?
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ANSWER KEY PAGE 171 103

 TABLE OF
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CONTENTS

This is a summary of
KNOWLEDGE SUMMARY you need to REMEMBER
what
. Fill
in the missing informa
tion.

Rational Expressions
P
A rational expression may be written in the form __
​ ,​​ where P and Q are ,
Q
but where Q  0.

Restrictions
A rational expression is properly defined only when the divisor is different from 0. The values that the
variables of this expression may not take are called the .

Example:
​x​2​2 1
In the rational expression ​___________​​ :
x(2x 2 5)(x 1 2)
• the denominator is x(2x 2 5)(x 1 2), so
This book is the property of Dickson Joseph.

x0 (2x 2 5)  0 (x 1 2)  0
The restrictions are therefore .

Adding and Subtracting Algebraic Expressions

To add and subtract rational expressions, you must follow the procedure below.
• Determine the .
• Transform all fractions into equivalent fractions with the same common denominator.
• Add or subtract the in the numerator position.
• Simplify the expression, if applicable.

Example:

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3n 2 1 ______
_____ 4 2 (3n 2 1)(n 2 4) 2 4 1 2(n)
​ ​2 ​ 2 ​1 ____
​ ​ 5 __________________
​   ​
n ​n​ ​2 4n n 2 4 ​n​2​2 4n
3​n​2​2 12n 2 n 1 4 2 4 1 2n
5 ​__________________
​     ​
​n​2​2 4n

104 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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Special Algebraic Identities

These are special equations that have been validated for all possible values that may be assigned to the
variables they contain.

Examples:
Square of a binomial: (a 1 b)2 5

Difference of two squares: a2 2 b2 5

Factoring Algebraic Expressions Using Identities

1. Perfect Square Trinomials
A perfect square trinomial is recognized by the fact that it is made up of two terms that are squares of

knowledge summary
monomials and a term that is the of these two monomials. A perfect square

chapter 2
trinomial can be factored using the square of a binomial.

a2 1 2ab 1 b2 5

Example:
Here is how to factor 9x4 2 12x2y 1 4y2 .
9x4 2 12x2y 1 4y2 5 (3x2 2 2y)2

The square root of This term equals The square root of

the 1st term is the double of the the 3rd term is
product of 3x2 and
. .
22y.

The two factors of 9x4 2 12x2y 1 4y2 are and

which, by convention, is written (3x2 2 2y)2.
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KNOWLEDGE SUMMARY PAGE 128 105

 TABLE OF
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2. Difference of Two Squares

A binomial that is the difference of two squares can be factored into the product of .
One of the binomials is the sum of the square roots of the two terms and the second is the difference
of the square roots of these same terms. The order is not important.

a2 2 b2 5 (a 2 b)

Example:
25x4 2 4y2 5 (5x2 1 2y)(5x2 2 2y)

The square root The square root Sum of the Difference

of the 1st term is of the 2nd term is square of the
roots square roots
. .

The two factors of 25x4 2 4y2 are and .

3. Simplifying Rational Expressions

This book is the property of Dickson Joseph.

By factoring, (simple factorization, perfect square trinomial, difference of two squares), it is possible to
simplify rational expressions.

Examples:
10x 2 5 ___________
______ 5(2x 2 1) 5 1 1
​ 2 ​5 ​ ​5 _____
​ ​ __
​, where x  2 ​ ​​and x  __
​​
4​x​ ​2 1 (2x 1 1)(2x 2 1) 2x 1 1 2 2

4​x​2​2 12x 1 9 ___________

__________ ​(2x 2 3)​​2​ 2x 2 3 3 3
​ ​5 ​ ​5 _____
​ ​, where x 2__
​ ​​and x  __
​​
4​x​ ​2 9
2 (2x 1 3)(2x 2 3) 2x 1 3 2 2

Multiplying Two Polynomials

To multiply two polynomials, simply apply the distributive property by multiplying each term of the second

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polynomial by each term of the first.

Example:
The following example is one way to multiply two second-degree trinomials.

(x2 2 2x 1 3)(2x2 1 2x 1 7) 5 2x4 1 2x3 1 7x2 2 4x3 2 4x2 214x 1 6x2 1 6x 1 21

After simplifying similar terms, a fourth-degree polynomial is obtained:

.

106 CHAPTER 2 – Algebraic Expressions

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Dividing a Polynomial by a Binomial

There is an algorithm that can be used to divide a polynomial by a binomial and which gives the
. Below are two examples (without and with remainder).

Example 1: Example 2:
Without remainder With remainder
(3x 2 16x 2 12) 4 (x 2 6)
2
(x3 2 3x2 1 4x 2 11) 4 (x 2 3)
3x2 2 16x 2 12 x26 x3 2 3x2 1 4x 2 11 x23
2 (3x2 2 18x) 3x 1 2 2 (x3 2 3x2) x2 1 4 Quotient
2x 2 12 0x2 1 4x 2 11
2 (2x 2 12) 2 (4x 2 12)
0 1 Remainder
Thus, you get (3x2 2 16x 2 12) 4 (x 2 6) 5 3x 1 2,

knowledge summary
The result can therefore be written as follows:
which may also be written as: 1
(x3 2 3x2 1 4x 2 11) 4 (x 2 3) 5 (x2 1 4) 1 ______

chapter 2
​ ,​​
3x2 2 16x 2 12 5 (x 2 6)(3x 1 2) x23
which may also be written as:
x3 2 3x2 1 4x 2 11 5 (x 2 3)(x2 1 4) 1 1

NOTE: When the remainder is 0, you can say that the polynomial is divisible by the binomial. In this case, the
polynomial can be factored and the binomial is one of its factors.
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KNOWLEDGE SUMMARY PAGE 130 107

 TABLE OF
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INTEGRATION
1 After checking that these expressions cannot be simplified by factorization, perform the following divisions.
10​x​2​2 12 1 14x
____________ 9​x​2​2 16
a) ​​   ​​ b) ​​_______​
5x 2 3 4 1 3x

29 1 4​x​2​ 10​x​2​2 16 2 12x

_______
c) ​
​ ​​ d) ​​____________
   ​
2x 1 3 5x 1 4

2 You know the identity associated with the square of a binomial: (a 1 b)2 5 a2 1 2ab 1 b2. There are
This book is the property of Dickson Joseph.

other identities associated with the cube or fourth power of a binomial.

a) Determine these identities by performing the following multiplications. Be sure to correctly order the
terms of the simplified polynomials you get.
1) (a 1 b)3 5 (a 1 b)(a 1 b)2 5

2) (a 1 b)4 5 (a 1 b)(a 1 b)3 5

DID YOU KNOW?

The illustration on the right depicts “Yang Hui’s Triangle,” named after
the Chinese mathematician who invented it in the 13th century to

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determine the power of the sum of two terms. This number triangle
is easy to build.

Simply proceed row by row. On each side, to the left and right, are
ones. Each number on the inside is the sum of the two numbers
above it. Note that this is also known as “Pascal’s Triangle,” named
after the French mathematician Blaise Pascal, who used a similar
triangle in the 17th century to solve a famous probability problem.

b) Using Yang Hui’s (or Pascal’s) Triangle, write the identity associated with (a 1 b)5.

108 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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3 Simon has observed that the difference of the squares of two consecutive odd numbers always seems to
be divisible by 8.
52 2 32 5 25 2 9 5 16 5 8 3 2
112 2 92 5 121 2 81 5 40 5 8 3 5
You know that an odd number can be written as 2n 1 1, where n is a natural number. The odd number
that comes after is greater by 2. This can therefore be written as 2n 1 3.
Using these algebraic expressions, prove that Simon’s conjecture is true.

4 Camilla wants to share her caramel chocolate bar equally between

(n 1 1) people. The bar contains n rows of (2n 1 1) pieces. If there are
any extra pieces left, they will be assigned randomly.
a) What is the minimum number of pieces each person will receive?
How many pieces will be assigned randomly?

integration
chapter 2
b) Identify a way to check your answer. Explain your procedure.

5 Adding x to each side of a square garden measuring (7x 2 1) metres per

side creates a path. In all, this forms a square measuring (9x 2 1) metres on
each side. To determine the algebraic expression that represents the area 9x  1

of the path, simply perform the subtraction below. 7x  1

(9x 2 1)2 2 (7x 2 1)2

Determine the simplified expression representing the area of the path using
two different methods.
a) By first calculating the squares of the binomials.
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b) By using the fact that it is the difference of two squares.

ANSWER KEY PAGE 172 109

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6 Matilda’s room is square. Her brother Ethan’s room is rectangular. Ethan’s room measures 1 m more in
length and 1 m less in width than Matilda’s room. Whose room is bigger? Justify your answer.

7 Consider this series of images.

The first image shows a piece of cardboard measuring a units per side and from which a square
This book is the property of Dickson Joseph.

measuring b units per side has been removed. This piece of cardboard is cut into two rectangles (second
image). The smaller rectangle is then moved to the right to make one large rectangle (third image).
a) Consider the first image. What expression represents the area of this piece of cardboard?

b) What are the dimensions of the large rectangle in the third image?

c) What special identity does this series of images illustrate? Justify your answer.

Mathias has observed that _​​ 2 ​​ 2 _​​  3 ​​produces the same result as _​​ 2 ​​ 3 _​​  3 .​​ Likewise, _​​  3 ​​ 2 _​​  4 ​​produces the same
1 1 1 1 1 1
8

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result as _​​  3 ​​ 3 _​​  4 .​​
1 1

a) Name two other unitary fractions whose difference is equal to their product.

b) Complete the following identity, where n represents an integer greater than zero.
1
__ 1 1 1
​ ​​2 _____________
​ ​​5 __
​ ​​3 _______
​ ​
n n
c) Prove that it is in fact an identity.

110 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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9 The warehouse illustrated below is in the shape of a rectangular prism. It is used to store cube-shaped
boxes. The dimensions of the warehouse are indicated on the diagram (the variable x is in metres).
a) What is the maximum number of cube-shaped boxes, measuring (x 2 2) on each side, that can be
stored in the warehouse?

(6x2  8x  6)

(8x2  12x  8)
(15x2  34x  8)

integration
chapter 2
b) Is it possible to completely fill the warehouse with boxes? If your answer is no, provide the algebraic
expression that can be used to calculate the empty space.

( )
x
10 Consider the following complex rational expression: ​ __________
​ ​ .​​
1
1 2 ______
​ ​
x11
a) What restrictions must be applied to x so that the expression is properly defined?

b) Evaluate this expression when x 5 1, x 5 2 and x 5 3.

c) Show that this expression is simply equivalent to x 1 1.

© SOFAD / All Rights Reserved.

The formula __
​​  b ​​ 1 _​​  d ​​ 5 ___________
a c a3d1c3b
11 ​​  b3d ​can be used to calculate the sum of two fractions. Provide similar
formulas for the three operations below.
a c
a) __ a c a c
​ ​​2 __
​ ​ b) __
​ ​​3 __
​ ​ c) __
​ ​​4 __
​ ​
b d b d b d

ANSWER KEY PAGE 174 111

 +3
TABLE OF
x6 CONTENTS ANSWER KEY
6
x3 ÷
4
3x 3x2
6x 3 -7
LES 2x

+3
x6
x3 ÷
64 2
3x 3x
6
2x
x 3
-7

The Amateur Mentalists

This book is the property of Dickson Joseph.

After seeing the variety show, Armando and Monica, two math students, believe they have understood
the tricks used by the prodigy and the telepath. They are confident that they can also come up with
mathematical procedures that would make other students in their department believe they have psychic
powers. They decide to ask volunteers to pick a number between 1 and 9 then use a calculator to evaluate an
algebraic expression and give the answer as a fraction. Meanwhile, Armando and Monica will use “telepathy”
to read the students’ thoughts, identifying the number they picked as soon as they complete the procedure.
Armando proposes a mathematical procedure that is represented by the following algebraic expression:
6​x​3​1 3​x​2​1 5x 1 1
_____________
​   ​1 (2​x​2​2 3x 1 2)(3​x​2​1 2x 2 1) 2 ​x​2​(3x 2 1)(2x 2 1) 2 7x​
2x 1 1
However, Monica thinks Armando’s method is too complex and proposes the following:
1) Pick a number from 1 to 9.
2) Multiply this number by itself and then by 3.

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3) Subtract 3 from the product and take note of this first result.
4) Go back to the initial number and multiply it by 6.
5) Add 3 to the product.
6) Multiply this sum by 1 more than than the initial number and take note of this second result.
7) Divide the first result by the second.
To test their methods, the two students ask a friend to try them out and to tell them which one is better. Their
friend says that, mathematically, there is no difference between the two. Is he right?
TASK

Your task is to prove, algebraically, whether their friend is right and to interpret the final answers
produced by Monica and Armando’s methods to determine the number chosen by the volunteer.

112 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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Solution

chapter 2
LES
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Proposition:

Assessment by criterion
Cr. 1.1 A B C D E
Cr. 1.2 A B C D E
Cr. 2.1 A B C D E
Cr. 2.2 A B C D E
Cr. 2.3 A B C D E

ANSWER KEY PAGE 175 113

 COMPLEMENTS

REFRESHER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

KNOWLEDGE SUMMARY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

MATHEMATICAL REFERENCE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

GLOSSARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

ANSWER KEY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

This book is the property of Dickson Joseph.

EVALUATION GRID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

 TABLE OF
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REFRESHER
REMINDER, PAGE 6 Concept of Functions

1 In each of the following situations, determine the dependent variable and the independent variable.
a) The number of words in a text and the time it takes to write them.
b) The surface area to be painted and the number of paint cans to be bought.
c) The average speed of a vehicle and the time it takes to make the journey.

2 Look at the graphs below. Which one(s) do(es) not represent a function?
Explain why.
A) B) C)
y y y
5 5 5
4 4 4
3 3 3
2 2 2
1 1 1
0 0 0
2 5 24 2 3 2 2 21 1 2 3 4 5 x 5 24 23 2221
2 1 2 3 4 5 x 5 24 23 22 21
2 1 2 3 4 5 x
22 22 22
23 23 23
24 24 24
25 25 25

3 In Martha’s kitchen, the tap flows at a rate of 120 ml/s. It takes 3 s to fill a 360 ml glass to the brim. Let t be
the time (in seconds) required to fill the glass and f(t), the amount of water it contains (in millilitres).

refresher
a) Determine the function rule f.
b) Draw a graph for this function.
c) What is the image of 2 under this function?
1
d) Calculate: f​​ __ ()
​ ​ ​​ and f(1.5).
2
e) Taking the context into account, what is the domain of this function?
f ) What is the range (codomain) of f ?
© SOFAD / All Rights Reserved.

g) How long would it take to fill the glass with:

1) 150 ml of water?
2) 300 ml of water?

4 Using the function f(x) 5 4x 2 6, evaluate the following expressions.

a) f(2) b) f(1.5) c) f(23) d) f(0)

ANSWER KEY PAGE 177 115

 TABLE OF
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REMINDER, PAGE 7 Intervals and Real Numbers

5 Fill in the table below so that each interval is represented three ways: in interval notation, in set-builder
notation and using a number line.

Interval Notation Set-builder Notation Number Line

a) [2, 5[ 3
2 2
2 2 1 0 1 2 3 4 5 R

b) {x  R | 0  x  5} 3
2 2
2 2 1 0 1 2 3 4 5 R

c) 3
2 2
2 2 1 0 1 2 3 4 5 R

d) ]21, 3[ 3
2 2
2 2 1 0 1 2 3 4 5 R

e) {x  R | x  0} 3
2 2
2 2 1 0 1 2 3 4 5 R

f) 3
2 2
2 2 1 0 1 2 3 4 5 R
This book is the property of Dickson Joseph.

g) ]2∞, 2] 3
2 2
2 2 1 0 1 2 3 4 5 R

h) {x  R | x  2} 3
2 2
2 2 1 0 1 2 3 4 5 R

6 Fill in the following table, assuming that the Universe is R.

Algebraic
Number Line Interval Notation
Representation

a) x  8

b) 2 R

c) [22, 3[

© SOFAD / All Rights Reserved.

d) 1
]2∞, 2​__
​  ​​]
2

e) 23 x0

f) ]2∞, 4]  ]10, 1∞[

g) 2 3 R

h) 4  x  6

116 REFRESHER
 TABLE OF
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7 Convert the following number sets into set-builder notation or interval notation.
a) {x  R | x  5} b) {x  R | 2  x  6}
c) ]2, 1∞[ d) [25, 4]

8 Which of the following numbers are included in the intervals provided?

A) 0 in the interval ]0, 1∞[ B) 24 in the interval [22, 8]
C) 22 in the interval ]25, 5[ D) 8 in the interval ]2∞, 10[

9 Determine whether the following statements are strictly positive, positive, negative or strictly negative.
a) x  0 b) x  0
c) x  0 d) x  0

10 Determine for which values of x the function is:

a) strictly positive or strictly negative. b) positive or strictly negative.

y y

6 6
5 5
4 4
3 3
2 2
1 1

0 0
2 6 25 24 23 22 21 1 2 3 4 5 6 x 2 6 25 24 23 22 21 1 2 3 4 5 6 x
22 22

refresher
23 23

24 24

25 25

26 26

c) negative or strictly positive. d) strictly positive or strictly negative.

y y

6 6
5 5
© SOFAD / All Rights Reserved.

4 4
3 3
2 2
1 1

0 0
2 6 25 24 23 22 21 1 2 3 4 5 6 x 6 25 24 23 22 21
2 1 2 3 4 5 6 x
22 22

23 23

24 24

25 25

26 26

ANSWER KEY PAGE 178 117

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REMINDER, PAGE 27 Rounding

11 Round the following numbers to the nearest integer.

a) 9.839 b) 0.3685 c) 7.979 d) 0.539 e) 967.4 f ) 1279.67

REMINDER, PAGE 67 Operations on fractions

12 The figure to the right represents the values

associated with each part of the Eye of Horus,
an ancient Egyptian symbol.
What is missing from the Eye of Horus in order 1
to be worth a full unit? 8

1
4
1 1
16 2
This book is the property of Dickson Joseph.

1
1 64
32

13 Carry out the following operations. Simplify the result if possible.

__ 1 1 1 1 1 1 1 1
a) ​​   ​​ 1 ___
​​   ​​ 5 b) __
​​   ​​ 2 ___
​​   ​​ 5 c) __
​​   ​​ 3 ___
​​   ​​ 5 d) __
​​   ​​ 4 ___
​​   ​​ 5
4 20 4 20 4 20 4 20

__ 5 8 5 8 5 8 5 8
e) ​​   ​​ 1 ___
​​   ​​ 5 f) __
​​   ​​ 2 ___
​​   ​​ 5 g) __
​​   ​​ 3 ___
​​   ​​ 5 h) __
​​   ​​ 4 ___
​​   ​​ 5
6 15 6 15 6 15 6 15

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REMINDER, PAGE 69 Simplifying Fractions

14 Using the greatest common denominator (GCD), simplify the following fractions to their simplest expression.

15
a) ___ 28
b) ___ 72
c) ___ 8
d) ___ 32
e) ___
​ ​ ​ ​ ​ ​ ​ ​ ​ ​
90 63 16 36 18

118 REFRESHER
 TABLE OF
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15 Make the following fractions irreducible.

12
a) ___ 50
b) ___ 14
c) ___ 48
d) ___ 44
e) ___
​ ​ ​ ​ ​ ​ ​ ​ ​ ​
15 60 84 90 28

33
___ 84
____ 81
___ 105
____ 60
____
f ) ​55 ​ g) ​105 ​ h) ​21 ​ i) ​ ​
267 j) ​168 ​

REMINDER, PAGE 69 Laws of Exponents

16 Perform the following multiplications in your head.

a) (4x)(1.5x) b) (x)(4x2) c) (3x3)2

1
d) (4x2)(3y) e) (6xy)(​​ __ ​​y) f ) (22x2)(2xy2)
2

17 Perform the following divisions.

(The result may not necessarily be a monomial.)
a) 6x 4 2x b) 8x3 4 5x c) 10x 4 5x2

d) 3xy 4 3x e) xy2 4 2xy f ) 4x2y 4 (22x)

refresher
REMINDER, PAGE 74 Product of Binomials

18 Perform the following multiplications.

a) (x 1 6)(2x 2 1) b) (5x 1 3)(3y 1 5) c) (6x 2 5)(6x 1 5)

19 Determine the products of the binomials. Simplify your answers.

a) (2x 2 5)(x 2 3) 5 b) (x 1 6)(3x 1 2) 5 c) (3x 2 4)(x 1 8) 5 d) (5x 1 3)(5x 2 3) 5
© SOFAD / All Rights Reserved.

REMINDER, PAGE 81 Simple Factorization

20 Break down the following polynomials into a product of factors using factorization.
a) 8x2 1 12x 5 b) 6xy 2 8x 5 c) 10x3 1 5x2 5

d) 12x2y 2 21xy2 5 e) 15x3 2 30ax2 1 9bx2 5

ANSWER KEY PAGE 180 119

 TABLE OF
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REMINDER, PAGE 88 Order of Operations

21 Solve the following series of operations.

a) 78.56 1 (2.9 3 17.23 2 1.1) 5 b) (13.2 2 10.34)(6.2 2 10) 5
c) 34.2 1 (21.44 4 3.2 2 5.1) 2 (6.1 2 12.1) 5 d) 1.1 2 (4.5 1 2.3(3.4 2 (5.2 2 2.6))) 5
e) 2.1 4 2.1 2 2.1(2.1 2 2.1(2.1 2 2.1) 2 2.1) 5

22 Solve the following series of operations.

a) (0.09 4 12) 3 3 2 4(2 2 5) 5 b) 5000 3 (8% 4 12) 3 3 2 3(22 2 8) 5
c) 2000 1 (2000 3 0.8 3 3) 5 d) 0.01((200 2 50)(24) 2 (25)(24) 2 20(6 1 4)) 5

REMINDER, PAGE 89 Terms and Degrees of Polynomials

23 Determine the degree of the following monomials.

This book is the property of Dickson Joseph.

a) 3x b) 24x2 c) 5xy2 d) 26

24 How can you determine the degree of a polynomial with more than one term?
As an example, determine the degree of the polynomial 5x2 2 4xy 1 3xy2.

REMINDER, PAGE 91 Algebraic and Arithmetic Division

25 Perform the following divisions.

a) (15x2y 2 5xy) 4 5xy b) (8x2y 2 2x3) 4 4x2
c) (15ax2 2 10ax) 4 5ax d) (6x3 1 12x2 2 3x) 4 (23x)

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REMINDER, PAGE 92 Dividing Numbers Using an Algorithm

26 Perform the following divisions.

a) 2952 24 b) 5525 17 c) 5024 32

120 REFRESHER
 TABLE OF
CONTENTS

KNOWLEDGE SUMMARY
CHAPTER 1
Step Functions
A step function is a function that is constant on each of its defining intervals and that jumps from one interval
to the next as the independent variable changes.
The critical values are the endpoints of the intervals where the function varies abruptly.

knowledge summary
As a result, the graph of the function is made up exclusively of horizontal segments called steps. A closed circle

chapter 1
() at the end of a step means that the endpoint is included in the graph of the function. An open circle ()
means the opposite. The image of a critical value always corresponds to the y-coordinate of the closed circle.

Example:
Graph of a Step Function Table of Values • The critical values of the
y function are 21 and 1.
5 • The closed endpoints
4 x f(x)
3
associated with these critical
2
[23, 21[ 3 values mean that f(21) 5 22
1
[21, 1[ 22 and f(1) 5 2.
25 4 3 2 1
2 2 2 2 1 2 3 4 5 x
22
[1, 4] 2
23

knowledge summary
24

25

chapter 1
Greatest Integer Function f(x) 5 [x]
Greatest integer functions are specific cases of step functions. The integer part of a number, written x,
is the greatest integer less than or equal to that number.
In the graph of a greatest integer function, all the steps are of the same length, and the jumps between
consecutive steps are equal in height. Step functions are also called staircase functions because of their
obvious resemblance to a staricase.
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The rule for this greatest integer function is written as follows: f(x) 5 x.

Example:
__ __
f(2.3) 5 2.3 5 2 ; f(​​√ 2
 ​​) 5 ​​√ 2
 ​​ 5 1 ; f(213.5) 5 213.5 5 214.

121
 TABLE OF
CONTENTS

Greatest Integer Function f(x) 5 a[bx]

In a greatest integer function of the form f(x) 5 abx:
• The first closed endpoint is at (0, 0)
• Parameter a determines the height of a jump; this height is equal to |a|
1
• Parameter b determines the length of a step; this length is equal to ___
​​   ​​.
|b|

Example:

Graph of a Greatest Integer Function Table of Values Height of one jump: 4

y Length of one step: 3

8
7 x f(x)
6 Value of parameter a: 4
5 [26, 23[ 28
1
4
Value of parameter b: __ ​​   ​​
3
[23, 0[ 24 3

[ ]
Closed 2 Jump
1
endpoint 1 Rule: f(x) 5 4​​ __
​   ​  x ​​
[0, 3[ 0 3
This book is the property of Dickson Joseph.

27 26 25 24 23 22 21 1 2 3 4 5 6 7 8 9 x
22

23
[3, 6[ 4
24

25 [6, 9[ 8
26
Step 27

28

The Values of a Greatest Integer Function

To determine a value of x when f(x) is known, follow these steps:
1) Isolate the integer part.
2) Determine the interval that satisfies the integer part. This interval always takes the form z, z 1 1,
where z is an integer.
3) Isolate x in the expression contained in the integer part.

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To determine a value f(x) when x is known, simply replace x in the rule with its value and follow the order of
operations. In this case, the integer part symbol has the same priority as brackets.

Example:
Let the function f(x) 5 230.25x.
If the value of x is 11, then: If the value of f(x) is 9, then: If the value of f(x) is 27, then:
f(11) 5 230.25 × 11 9 5 230.25x 2 7 5 230.25x
f(11) 5 232.75 3 5 0.25x
2 2,3 5 0.25x
f(11) 5 23 3 2 5 26 3  0.25x , 22
2 Since no number has an integer part
of 2,3, there is no solution.
2 12  x , 28

122 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

Greatest Integer Function f(x) 5 a[b(x 2 h)] 1 k

The Effect of Changing Parameters a, b, h and k
• Reversing the sign of parameter a produces a reflection over the x-axis.
• Reversing the sign of parameter b produces a reflection over the y-axis.

yy yy

11 11 If parameter b If parameter b
is positive is negative
11 xx 11 xx

Effect of reversing the Effect of reversing the

sign of parameter a sign of parameter b

• Increasing |a| increases the height of the jump. The function is said to undergo a vertical stretch.
Decreasing |a| has the opposite effect: a vertical compression.
• Increasing |b| decreases the length of the step. The function is said to undergo a horizontal compression.
Decreasing |b| has the opposite effect: a horizontal stretch.
• Increasing parameter h results in a horizontal translation to the right. Decreasing h has the opposite effect:

knowledge summary
a horizontal translation to the left.

chapter 1
• Increasing parameter k results in an upward vertical translation. Decreasing k has the opposite effect: a
downward vertical translation.

Example:
In the function f(x) 5 (x – 4) 1 5, the function undergoes a horizontal translation of 4 units to the right
and a vertical translation 5 units upwards. Consequently, there is a closed endpoint at (4, 5).
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123
 TABLE OF
CONTENTS

Graphing Function f(x) 5 a[b(x 2 h)] 1 k

To graph a greatest integer function, you must determine the following information from the parameters
of the standard form.
• The coordinates of a closed endpoint: (h, k).
1
• The length of one step: ______
​​   ​​.
|​ b |​
• Whether the closed endpoints are on the left (b . 0) or the right (b , 0).
• Whether the function is increasing (a and b have the same sign) or decreasing
(a and b have opposite signs).
• The height of the jumps: |a|.

From the Graph to the Rule of a Greatest Integer function

To determine the rule for a greatest integer function in its standard form, f(x) 5 ab(x − h) 1 k,
you must find the values of parameters a, b, h and k by following these steps.
1) Determine possible values for parameters h and k by selecting a closed endpoint on the graph.
2) Determine |a| from the height of the jumps.
This book is the property of Dickson Joseph.

3) Determine |b| from the inverse of the length of a step.

4) Determine the sign of b from the direction of the closed endpoints.
5) Determine the sign of a by noting whether the function is increasing or decreasing and by taking
into account the sign of parameter b.

Example:
Find the corresponding rule for the following graph:
y

1) Possible values of parameters h and k are h 5 2 and k 5 0.

2 24

2) |a| 5 4 16

1
3) |b| 5 __
​​   ​​ 8

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2
4) According to the direction of the endpoints, b is positive. 2 8 6
2 4
2 4 8 12 x
1
Therefore, b 5 __
​ ​  ​​. 8
2

2
16
2
5) Since the function is decreasing and parameter b is positive,
parameter a will be negative. Therefore, a 5 24. 24
2

Consequently, the function rule is f(x) 5 24​​ __

1
[ ]
​   ​  (​x 1 2) .​​
2

124 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

From the Table of Values to the Rule of a Greatest Integer Function

Follow these steps to determine the rule for a greatest integer function of the form
f(x) 5 ab(x 2 h) 1 k using a table of values.

1) Determine possible values of parameters h and k. In the table of values, parameter h is the value of the
closed side of the interval and parameter k is the value of the associated independent variable.
2) Determine |a| from the height of the jumps. In the table of values, the height of the jumps is the difference
between two consecutive values of the dependent variable.
3) Determine |b| from the inverse of the length of a step. In the table of values, the length of the steps is the
difference between the endpoints of each interval.
4) Determine the sign of b by noting which side of the interval is closed. If the interval is closed on the left,
b is positive; if the interval is closed on the right, b is negative.
5) Determine the sign of a by noting whether the function is increasing or decreasing and by taking into
account the sign of parameter b.

Example:
The steps below show how to determine the corresponding rule for the following table of values.

x y
[26, 21[ 12
difference of 3

knowledge summary
[21, 4[ 9

chapter 1
difference of 3
[4, 9[ 6
difference of 3
[9, 14[ 3

1) The endpoint (h, k) 5 (4, 6) is closed because the interval [4, 9[ is closed at 4 and the value of
the y-coordinate at this point is 6. Therefore, two possible values of parameters h and k are h 5 4
and k 5 6.
2) The difference between two consecutive values of the dependent variable is 3 (e.g. 12 2 9 5 3),
so the jumps have a height of 3 units and |a| 5 3.
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3) The difference between the endpoints of the intervals is 5 (e.g. 21 2 (26) 5 5), so the steps are
1
5 units long and |b| 5 __
​​   ​​.
5

4) The intervals are closed on the left, so the closed endpoints are at the beginning of the steps
and parameter b is positive.
5) When the values for x increase, the values for y decrease, so the function is decreasing.
A decreasing function means that a and b have opposite signs. Since parameter b is positive,
parameter a must be negative.

The function rule is therefore f(x) 5 23​​ __

1
[ ]
​   ​  (​x 2 4) ​​ 1 6.
5

125
 TABLE OF
CONTENTS

Function Properties Applied to the Greatest Integer Function

Example:
The graph to the right represents y
the function f(x) 5 2[2x]. f(x)  2[2x]
6

4
Images

Zeros

3 2 1
 1 2 3 x

2
 y-intercept

4


The function is negative. The function is positive.

This book is the property of Dickson Joseph.

Domain

Property Definition Example

Domain The domain of a function is the set of values that the Set of real numbers: R
independent variable may have.
Range (codomain) The range (or codomain) of a function is the set of {… 26, 24, 22, 0, 2, 4, 6 …}
values that the dependent variable may have.
Zeros of the function The zeros of the function are the values for which [0, 0.5[
(x-intercepts) f(x) 5 0. There may be none, one or several.
y-intercept The initial value is the value of f(0). In specific contexts, 0
it is often called the start value. In a graph, it is called Out of context, the term
the y-intercept. y-intercept is more common.
Sign The sign of a function may be positive or negative. Negative over x  ]2∞, 0]

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It is expressed by the intervals of the domain of the Positive over x  [0, 2∞[
function where f(x)  0 if the function is positive
and where f(x)  0 if the function is negative.
Change The direction of change indicates whether the Increasing over the
function is increasing or decreasing. entire domain
Extrema The extrema are maximum or minimum values, No extrema
if they exist.

126 CHAPTER 1 – Step Functions and Greatest Integer Functions

 CHAPTER 2
KNOWLEDGE SUMMARY TABLE OF
CONTENTS

Rational Expressions
P
A rational expression can be written in the form __
​ ,​​ where P and Q are polynomials, but where Q  0.
Q

Restrictions
A rational expression is properly defined only when the divisor is different from 0. The values that the
variables of this expression may not take are called the restrictions.

knowledge summary
chapter 2
Example:
​x​2​2 1
In the rational expression ​___________​​ :
x(2x 2 5)(x 1 2)
• the denominator is x(2x 2 5)(x 1 2), so

x0 (2x 2 5)  0 (x 1 2)  0
5
The restrictions are therefore x  0, x  __
​ ​​and x  22.
2

Adding and Subtracting Algebraic Expressions

To add and subtract rational expressions, you must:
• determine the common denominator;

knowledge summary
• transform all fractions into equivalent fractions with the same common denominator;

chapter 2
• add or subtract the similar terms in the numerator position;
• simplify the expression, if applicable.

Example:
3n 2 1 4 2 (3n 2 1)(n 2 4) 2 4 1 2(n)
​_____​2 ______
​2 ​1 ____
​ ​ 5 ​__________________
  ​
n ​n​ ​2 4n n 2 4 ​n​2​2 4n
3​n​2​2 12n 2 n 1 4 2 4 1 2n
5 ​​__________________
    ​
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​n​2​2 4n
3​n​2​2 11n
5 ​​________ ​
​n​2​2 4n
n(3n 2 11)
5 ​​________​
n(n 2 4)
3n 2 11
5 ______
​ ,​​ where n  0 and n  4.
n24

127
 TABLE OF
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Special Algebraic Identities

These are special equations that have been validated for all possible values that may be assigned to the
variables they contain.

Examples:
Square of a binomial: (a 1 b)2 5 a2 1 2ab 1 b2

Difference of two squares: a2 2 b2 5 (a 2 b)(a 1 b)

Factoring Algebraic Expressions Using Identities

1. Perfect Square Trinomials
A perfect square trinomial is recognized by the fact that it is made up of two terms that are squares of
monomials and a term that is the double of the product of these two monomials. A perfect square trinomial
can be factored using the square of a binomial.

a2 1 2ab 1 b2 5 (a 1 b)2
This book is the property of Dickson Joseph.

Example:
Here is how to factor 9x4 2 12x2y 1 4y2 .
9x4 2 12x2y 1 4y2 5 (3x2 2 2y)2

The square root of This term equals The square root of

the 1st term is 3x2. the double of the the 3rd term is 22y.
product of 3x2
and 22y.

The two factors of 9x4 2 12x2y 1 4y2 are (3x2 2 2y) and (3x2 2 2y) which, by convention, is written (3x2 2 2y)2.

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128 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

2. Difference of Two Squares

A binomial that is the difference of two squares can be factored into the product of two binomials. One of the
binomials is the sum of the square roots of the two terms and the second is the difference of the square roots
of these same terms. The order is not important.

a2 2 b2 5 (a 2 b)(a 1 b)

Example:
25x4 2 4y2 5 (5x2 1 2y)(5x2 2 2y)

The square root The square root Sum of Difference

of the 1st term of the 2nd term square of square
is 5x2. is 2y. roots roots

The two factors of 25x4 2 4y2 are (5x2 1 2y) and (5x2 2 2y).

3. Simplifying Rational Expressions

By factoring (simple factorization, perfect square trinomial, difference of two squares), it is possible to simplify
rational expressions.

Examples:

knowledge summary
10x 2 5 ___________
______ 5(2x 2 1) 5 1 1
​5 _____​, where x  2​ __
​ ​​and x  __

chapter 2
​ 2 ​5 ​ ​ ​​
4​x​ ​2 1 (2x 1 1)(2x 2 1) 2x 1 1 2 2

4​x​2​2 12x 1 9 ___________

​(2x 2 3)​​2​ 2x 2 3 3 3
​__________ ​5 ​ ​5 _____
​ ​, where x  ​2 __
​ ​​and x  __
​​
4​x​ ​2 9
2
(2x 1 3)(2x 2 3) 2x 1 3 2 2

Multiplying Two Polynomials

To multiply two polynomials, simply apply the distributive property by multiplying each term of the second
© SOFAD / All Rights Reserved.

polynomial by each term of the first.

Example:
The following example is one way to multiply two second-degree trinomials.

(x2 2 2x 1 3)(2x2 1 2x 1 7) 5 2x4 1 2x3 1 7x2 2 4x3 2 4x2 214x 1 6x2 1 6x 1 21

After simplifying similar terms, a fourth-degree polynomial is obtained:

2x4 2 2x3 1 9x2 2 8x 1 21.

129
 TABLE OF
CONTENTS

Dividing a Polynomial by a Binomial

There is an algorithm that can be used to divide a polynomial by a binomial and which gives the quotient and
the remainder. Below are two examples (without and with remainder).
Example 1: Example 2:
Without remainder With remainder
(3x 2 16x 2 12) 4 (x 2 6)
2
(x3 2 3x2 1 4x 2 11) 4 (x 2 3)
3x2 2 16x 2 12 x26 x3 2 3x2 1 4x 2 11 x23
2 (3x2 2 18x) 3x 1 2 2 (x3 2 3x2) x2 1 4 Quotient
2x 2 12 0x2 1 4x 2 11
2 (2x 2 12) 2 (4x 2 12)
0 1 Remainder
So we get (3x2 2 16x 2 12) 4 (x 2 6) 5 3x 1 2,
The result can therefore be written as follows:
which may also be written as: 1
(x3 2 3x2 1 4x 2 11) 4 (x 2 3) 5 (x2 1 4) 1 ______
​ ,​​
3x2 2 16x 2 12 5 (x 2 6)(3x 1 2) x23
which may also be written as:
x3 2 3x2 1 4x 2 11 5 (x 2 3)(x2 1 4) 1 1
This book is the property of Dickson Joseph.

NOTE: When the remainder is 0, we say that the polynomial is divisible by the binomial. In this case, the polynomial can be
factored and the binomial is one of its factors.

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130 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

MATHEMATICAL REFERENCE
Mathematical Symbols
Symbol Meaning Symbol Meaning
5 … equals …  Intersection of sets

< … is approximately equal to … ∞ Infinity

 … not equal to … 1∞ Positive infinity

 Plus or minus 2∞ Negative infinity

 … less than … R Set of real numbers

 … greater than … R1 Set of positive real numbers

 … less than or equal to … R2 Set of negative real numbers

 … greater than or equal to … x Change in the x coordinate

{l} Empty set y Change in the y coordinate

 Empty set m Slope

[a] Integer part where a  R mAB Slope of the line AB

[a, b] Interval of a to b inclusive || Parallel

[a, b[ Interval including a, but excluding b ^ Perpendicular

]a, b] Interval excluding a, but including b |a| Absolute value of a

]a, b[ Interval of a to b exclusive dom f Domain of the function f

 … belongs to … img f Image of the function f

MATHEMATICAL
… does not belong to … codom f Codomain of the function f

REFERENCE

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 Union of sets

131
 TABLE OF
CONTENTS

Units of Measurement
Quantity: Time
L litre(s) s second(s)

ml millilitre(s) min minute(s)

Measurement d day(s)
mm millimetre(s)
h hour(s)
cm centimetre(s)
y year(s)
dm decimetre(s)
Other
m metre(s) % percentage

km kilometre(s) ° degree

cm2 square centimetre(s) °C degrees Celsius

m2 square metre(s) °F degrees Fahrenheit

This book is the property of Dickson Joseph.

km/h kilometre(s) per hour ¢ cent

m/s metre(s) per second $ dollar

m/s2 metre(s) per second squared GB gigabyte(s)

m/min metre(s) per minute kilowatt(s)

kW
(unit of electric power)
L/(100 km) litre(s) per 100 kilometres kilowatt hour(s)
kW h
(unit of energy use)
Weight
¢/(kW h) cent(s) per kilowatt hour
g gram(s)

kg kilogram(s)

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132 MATHEMATICAL REFERENCE

 TABLE OF
CONTENTS

Formulas Exponential Notation

Area formulas Illustration (figure) Notation Example
Area of a square an 5 a 3 a 3 … 3 a 25 5 2 3 2 3 2 3 2 3 2 5 32
n times 41 5 4
Asquare 5 s2 s a0 5 1 1250 5 1

1 1
a2n 5 __ 324 5 __ ​ 4 ​​5 ___
1
​​  ​a​​​ n​ ​​​ ​ ​
​3​ ​ 81
Area of a rectangle _1 __ _1
___
​​a​​ ​​ ​ 2​​​ 5 ​√ a ​ ​​81​​  ​ 2​​​ 5 ​√ 81 ​5 9

_1 __ _1
___
3
​​a​​ ​​ ​ 3​​​ 5 √
​3 a ​ ​​64​​  ​ 3​​​ 5 √
​ 64 ​​5 4
Arectangle 5 b 3 h h
___ ____ ____
__ _2 3
​​27​​  ​ 3 ​​​ 5 √
m n 3
​​a​​ ​​ ​  n ​​​​​​ 5 √
​ ​a​​​ m​​​ ​ ​27​​2​​ 5 √
​ 729 ​​5 9

Note: The base a represents a real number (except 0), while

b
m and n represent natural numbers greater than 0.
Area of a triangle

b×h Laws of Exponents

Atriangle 5 _____
​ ​
2 h Law Example
Product of powers
23 3 24 5 23 1 4 5 27 5 128
am 3 an 5 am 1 n
b
Quotient of powers ​3​5​
__
​ ​ ​​5 35 2 1 5 34 5 81
​a​​​ m​​
___ ​3​
​ n​​​5 am 2 n
​a​​​ ​
Power of a product (3 3 7)2 5 32 3 72 5
(ab)m 5 ambm 9 3 49 5 441

Power of a power
(43)2 5 43 3 2 5 46 5 4096
(am)n 5 amn

MATHEMATICAL
REFERENCE
© SOFAD / All Rights Reserved.

Power of a quotient

()a m​​ ​a​​​ m​​

​ __
​ ​ ​ ​​5 ___
b
​ m​​
​b​​​ ​
()3 2 ​3​2​ ___
​ ​ ​ ​​5 __
​ __
8 ​8​ ​
9
​ 2 ​​5 ​ ​
64

Note: The bases a and b represent real numbers (except 0),

while m and n represent natural numbers greater than 0.

133
 TABLE OF
CONTENTS

GLOSSARY
Absorbing Element Coefficient of Proportionality
An absorbing element is a number that, by means of In a situation of direct proportionality, this is the
a certain operation, transforms all other numbers or value by which the independent variable must be
expressions into itself. multiplied in order to obtain the dependent
variable. (Also see Proportional.)
Example:
The absorbing element of multiplication is 0, since any Example:
number multiplied by 0 will always equal 0. The perimeter P of a square is directly proportional to
the measurement s of its side. The coefficient of
proportionality of this relationship is the value of the
Algorithm P
ratio __
​​   ​​ , which is equal to 4. Therefore P 5 4s.
s
A series of steps that, when followed, will produce
the desired result regardless of the initial data.
Completing the Square
Axis of Symmetry A calculation method used to complete a binomial
A straight line that splits a geometric shape into two in order to produce a perfect square trinomial.
isometric parts by reflection over the axis.
This book is the property of Dickson Joseph.

Example:
Binomial By adding 9 to the binomial x2 2 6x, we get the perfect
square trinomial x2 2 6x 1 9, the factored form of
A polynomial with two terms. which is (x 2 3)2.

Codomain
Conjecture
The set of values that the dependent variable of a
A statement that is accepted as true but has not yet
function may have. (Synonym: Range of a function.)
been proven.
Example:
Constant (function)
Temperature of a liquid
as a function of time A function whose dependent variable can be only
Temperature
(°C)
6 one value. The function rule for this function is

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5 f(x) 5 a, where parameter a is a real number. On a
4 graph, a constant function is represented by a
3
2
horizontal straight line.
Codomain 1
0
1
2 1 2 3 4 5 6 7 8 9 10
2
2 Time
(min)
3
2

The codomain of this function is [22, 4] °C.

134 GLOSSARY
 TABLE OF
CONTENTS

Critical Values Degree

In a step function, these are the values of the The degree of a monomial is the sum of the
domain where the function is discontinuous—that exponents of the constituent variables.
is, where there is a jump on the graph.
Example:
Example: The degree of 5a2b is 3.
In the graph below, the critical values are 3 and 6.
The degree of a polynomial corresponds to the
y highest degree of its terms.
10
9 Example:
8
In the polynomial 3ab2 1 2a3b4 − 10, the degree is 7.
7
6
5 Denominator
4
3 The expression located below the fraction bar.
2
1 Example:
In the fraction _​​  3 ,​​ 3 is the denominator.
2
0 1 2 3 4 5 6 7 8 9 10 x

Critical values
Dependent Variable
In a functional relationship between two variables,
Decreasing this is the variable whose variations must be
A function is said to be decreasing within a given explained by the variations of another variable.
interval when it has the following property:
if x1  x2, then f(x1)  f(x2). Direct Proportionality
On a graph, it is easy to see when a function is Two variables are said to be directly proportional
decreasing over an interval because the curve if their ratio is equal to a constant. It is possible
representing it goes downwards when scanning to recognize that two variables are directly
from left to right. proportional if multiplying the value of one variable
by a number has the same effect as multiplying the
Example:
value of the other variable by the same number.
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7 Example:
6
5
When an object moves at a constant speed, the
4 distance it has travelled is directly proportional to the
Decreasing
3 time that has passed. To travel twice the distance, it
2
takes twice the time.
1
glossary

22 1
2 1 2 3 4 5 6 7 x
2
2

135
 TABLE OF
CONTENTS

Discontinuous (function) Distributive Property

A function is discontinuous when there are breaks in The property of one operation on another, making it
its graphical representation. possible to calculate the value of an expression in
two different ways.
Example:
The step function below is discontinuous at critical
Example:
values 1, 3 and 5. The distributive property of multiplication on addition
makes it possible to change the product of a number
y
and a sum into a sum of products.
6
5
A 3 (B 1 C) 5 A 3 B 1 A 3 C
4 Factor A is said to be distributed to each term in the sum.
3
2
1 Divisibility
0 1 2 3 4 5 6 x A number or an expression is divisible by another
number or another expression when the remainder
of the division is zero.
Critical values
Example:
This book is the property of Dickson Joseph.

x3 2 8 is divisible by x 2 2, since
Discrete Value
(x3 2 8) 4 (x 2 2) 5 x2 1 2x 1 4.
A finite or countable number attributed to a
mathematical variable.
Domain
Example: The set of values that the independent variable of a
“My neighbour has 5 pets.” function may have.
The number 5 is a discrete value of the variable “the
Example:
number of pets my neighbour has.”
The domain of this function is ]28, 7].

Discriminant y
5
Expression equal to b2 2 4ac, calculated based on 4
3
the coefficients of a second-degree equation of the

© SOFAD / All Rights Reserved.

2
form ax2 1 bx 1 c 5 0. This expression is used to 1
determine the number of real roots in the equation.
2 8 27 26 25 24 3 22
2 21 1 2 3 4 5 6 7 x
22
Examples: 23

24
If b2 2 4ac  0, the equation has two distinct roots.
5
2

If b 2 4ac 5 0, the equation has one real root.

2

If b2 2 4ac  0, the equation has no real roots.

136 GLOSSARY
 TABLE OF
CONTENTS

Equation Greatest Integer

An equation is a mathematical statement containing The largest integer that is less than or equal to a real
variables and an equals sign. The solution set of an number. The expression [x] represents the greatest
equation is made up of the values of the variables integer of a number x.
that transform the equation into a true equality.
Example:
Example:
[1.56] 5 1 [p] 5 3 [21] 5 21 [21.1] 5 22
x 1 3 5 4 is an equation whose solution set contains
only the number 1. The greatest integer function f(x) 5 [x] is a step function.

Factored Form Greatest Integer Function

The writing of the function rule of a second-degree A function that associates any real number with its
polynomial in the form of a product: integer part. The function rule of the greatest
f(x) 5 a(x 2 x1)(x 2 x2). integer function is f(x) 5 [x].
The parameters x1 and x2 are the zeros of the function.
Example:
A polynomial function that has no zero cannot be
written in factored form. 9 . 685

Greatest Integer Fractional part

Factorization
The greatest integer function is represented by a series
A procedure that consists of writing an algebraic of steps measuring 1 unit in length. The height of the
expression in the form of a product. The purpose is jumps between the steps (that is, the distance
to simplify the expression in order to solve a between two successive segments) is also 1 unit.
problem more easily. y

5
4
Function 3
2
A relationship between two variables—one 1
dependent and the other independent—in which 0
25 24 23 22 1 2 3 4 5 x
any value of the independent variable is associated 22
with only one value of the dependent variable. 23

24

The rule of the function, which is generally 25

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expressed by an equation, is used to calculate the

value of the dependent variable that corresponds to
each possible value of the independent variable.

General Form
The writing of an equation or the rule for a function
glossary

using a simplified polynomial.

Examples:
1) The general form of the equation for a straight line
is Ax 1 By 1 C 5 0.

2) The general form of a second-degree polynomial

function is f(x) 5 ax2 1 bx 1 c.

137
 TABLE OF
CONTENTS

Half-plane Independent Variable

The part of the plane delimited by a line. The line The variable that affects the dependent variable.
may be included in the half-plane or excluded.
Inequality
Example:
y A mathematical statement that includes at least one
algebraic expression and a relationship of inequality.

Example:
3x 2 4  x 1 6 is an inequality.
x

Initial Value
The value of f(0). In some contexts, this is often
called the start value. (Synonym: y-intercept.)
Identity (algebraic)
Integer
An algebraic equation that is true regardless of the
values attributed to the variables it contains. A natural number to which a positive or negative
sign is applied. The set of integers is represented by
This book is the property of Dickson Joseph.

Special identities can be used in algebraic

the letter .
calculations to simplify expressions more quickly
or to factor polynomials. 5 {… 23, 22, 21, 0, 1, 2, 3 …}

Examples:
Intercept
1) Identity of the perfect square trinomial:
On a Cartesian plane, an intercept is a point where
(a 1 b) 2 5 a 2 1 2ab 1 b 2
a curve or straight line crosses one of the axes (x or
2) Identity of the difference of two squares: y axis).
(a 1 b)(a 2 b) 5 a 2 2 b 2
Example:
Increasing The curve below has three intercepts: two x-intercepts,
at 1 and 5, and one y-intercept, at 2.
A function is said to be increasing within a given
interval when it has the following property: y

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if x1  x2, then f(x1)  f(x2). 7
6
On a graph, it is easy to see when a function is increasing 5
y-intercepts
over an interval because the curve representing it goes 4
3
upwards when scanning from left to right. 2
1
Example:
y 22 1
2 1 2 3 4 5 6 7 x
2
2
7 x-intercepts
6
5
Increasing
4
3
2
1

2
2 1
2 1 2 3 4 5 6 7 x
2
2

138 GLOSSARY
 TABLE OF
CONTENTS

Interval Natural Number

The set of real numbers between two endpoints, A number that can be used to describe the quantity
which may or may not be included in the interval. of elements in a finite set. These numbers are used
for counting. The set of natural numbers is
Example: represented by the letter .
The interval of the numbers from 2 (included) to 5 5 {0, 1, 2, 3, …}
(excluded) may be represented as follows:

Graphically: Numerator
2 5
Algebraically: 2x5 The expression located above the fraction bar.
In interval notation: [2, 5[
Example:
4
An interval may extend to infinity, either positive or In the fraction __
​ ,​​ 4 is the numerator.
5
negative. In this case, there is only one endpoint.

Example: Order of Operations

The interval [3, 1∞[ has only a lower endpoint A mathematical convention specifying the order in
(included). which operations are to be carried out to solve a
numerical expression. According to this convention,
Maximum the order of priority is:

The greatest value (number M) that a function f may 1) operations in brackets;

have if both of the following conditions are met: 2) exponents;
• There is a number a in the domain of f, such that 3) divisions and multiplications;
f(a) 5 M
4) additions and subtractions.
• f(x)  M for any x in the domain of f.
Operations of the same level of priority are carried
out as they appear, from left to right.
Minimum
The least value (number m) that a function f may Parabola
have if both of the following conditions are met:
The characteristic curve of the graph of a second-
• There is a number a in the domain of f, such that degree polynomial function. This curve has an axis
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f(a) 5 m of symmetry that goes through the vertex of the

• f(x)  m for any x in the domain of f. parabola.

Example:
Monomial
Axis of
An algebraic expression that contains only one term. symmetry

This term may consist of a number, a variable or a

glossary

parameter, or a product of these.

Parabola
Example:
3, 2x2 and 24ax represent three monomials.
Vertex

139
 TABLE OF
CONTENTS

Parameter Proof
In an algebraic expression or an equation, this is a A proof demonstrates beyond all doubt that a
letter that represents a real number that can have statement is true, using reasoning that applies to
different values depending on the case, rather than all possible situations.
a variable.
Proportional to the Square
Example:
A variable y is proportional to the square of a
variable x, if the ratio __
In the standard form of the function rule f(x) 5 ax 1 b, ​x​2​
​​  y ​​ is constant. The variable y is
the letter x represents the independent variable, while recognized as proportional to the square of x if
the letters a and b are the parameters. multiplying the value of x by a number has the same
effect as multiplying the value of y by the square of
that number.
Perfect Square Trinomial
A perfect square trinomial is made up of two terms Example:
that are squares of monomials and a term that is the
double of the product of these two monomials. The total area of a cube is proportional to the square of
the length of its edge. If the length of the edge is
Example: doubled, the area of the cube will be multiplied by 4
(that is, 22).
The trinomial 16x2 2 40x 1 25 is a perfect square,
This book is the property of Dickson Joseph.

because 16x2 and 25 are the squares of 4x and 25,

and 240x is double the product of 4x and 25. Quadratic Formula
It can therefore be factored as follows:
16x2 2 40x 1 25 5 (4x 2 5)2. A formula that is used to find the x-intercepts, or
zeros, of a second-degree polynomial function.
_________
2b  √​ ​b​2​2 4ac ​
Polynomial x 5 ​_______________
   ​
2a
An algebraic expression resulting from the sum of
monomials.
Example:
Example: Consider the function f(x) 5 x2 2 5x 2 9.75, with the
parameters a 5 1, b 5 25 and c 5 29.75.
The expression x2 2 4x 1 5 is a polynomial. It is ________________ ___
the sum of three monomials x2, 24x and 5. 56√ ​ ​(  
25)​​2​2 4(1)(29.75) ​ 5 6 ​√ 64 ​ _____
568
x 5 ​____________________
     ​​5 ______ ​ ​​5 ​ ​
2(1) 2 2

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A monomial may be considered a single-term
The two x-intercepts are 21.5 and 6.5.
polynomial.

Polynomial Function
A function whose rule is expressed using a
polynomial. The degree of the polynomial
determines the degree of the function.

Examples:
1) 0 degree polynomial function: f(x) 5 3

2) 1st degree polynomial function: f(x) 5 2x 1 3

3) 2nd degree polynomial function: f(x) 5 x2 1 2x 1 3

140 GLOSSARY
 TABLE OF
CONTENTS

Quadratic Function Rational Expressions

A function whose rule is written in the form Rational expressions are generalizations of ordinary
f(x) 5 ax2, where the coefficient a is a real fractions. They may contain numbers and variables.
number other than 0. (Synonym: Algebraic fraction.)
In the Solutions series, this expression extends A rational expression is written in the form of a
fraction __
P
to all second-degree polynomial functions. ​​  Q ,​​ where P and Q are polynomials and
Q  0.
Example:
Graph of the quadratic function f(x) 5 x2. Example:
y
“A number squared minus eight, divided by this same
5 number plus two” may be translated into the rational
f
expression ____
​x​2​2 8
4 ​​  x 1 2 ​​, where x  22.
3
2
1
Real Number
0
24 23 2 2 1
2 1 2 3 4 x A number that represents a point on a number line.
2
2
Real numbers include natural numbers, integers,
rational numbers and irrational numbers.
Range (Range of a function or Image) The set of real numbers is represented by the
1) The set of values that the dependent variable of symbol .
a function may have. (Synonym: Codomain.)
2) The resulting value when a function is applied to Reflection
a number. A geometrical transformation defined by an axis of
reflection. The image of a reflected two-dimensional
Example:
shape corresponds to the image that would be
Let f(x) 5 2x 1 1. obtained if the plane were inverted, that is, pivoted
The image of 2 under the function f, written f(2), is around the axis of reflection.
equal to 5.
Example:

Rate of Change Axis of reflection

A Aˈ
The ratio between the change in values of the
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dependent variable and the change in values of the

independent variable.
The rate of change is calculated using the B C Cˈ Bˈ

expression ________
f(​x​ ​) 2 f(​x​1​)
​​  2
​x​2​2 ​x​1​ .​​
In the case of an affine function, the rate of change
glossary

corresponds to the slope of the straight line that

represents the function on a graph. (Synonym:
Slope of a straight line.)

141
 TABLE OF
CONTENTS

Restriction Similar Terms

In an expression, the values that the variables may In an algebraic expression, these are terms that have
not take are the restrictions. the same variables with the same respective
exponents. The only difference between the terms is
Example: their numerical coefficient.
​x​2​ 2 1
_________ Example:
​ ​
3y(x 2 1)
In the algebraic expression
In the rational expression above, the variables 3y and
2ab − 2a 1 3a2b − 4ab2 − 5a2b,
x 2 1 cannot be equal to 0 because that would make
the terms 3a2b and 5a2b are similar.
the denominator equal to 0. Therefore, the restrictions
are y  0 and x  1.
Slope
Root (of an equation) A number that describes the steepness of a straight
line in relation to the horizontal.
Each solution to a single-variable equation.
On a Cartesian plane, the slope of a line is equal to
Rounding Down the ratio of the change in the y coordinate (y) to
the change in the x coordinate (x), when moving
A procedure of approximation by which a number is from one point to another along this line.
This book is the property of Dickson Joseph.

replaced with the lower value that is closest to this

number in accordance with the level of precision The slope of the line going through the points
required. P1(x1, y1) and P2(x2, y2) is calculated using the
following formula:
y ​y​2​2 ​y​1​
Example: Slope ​5 ___​   ​ 5 _______
​ ​
x ​x​2​2 ​x​1​
The result of rounding down the number 3.65 to the
nearest integer is 3.
The slope of a horizontal line is 0. The slope of a
vertical line is not defined.

Rounding Up Example:
A procedure of approximation by which a number is y
replaced with the higher value that is closest to this
number in accordance with the level of precision
required. (x1, y1)

© SOFAD / All Rights Reserved.

y
(x2, y2)
Example: x
x
The result of rounding up the number 3.65 to the
nearest integer is 4.

Rule
A function rule, which is usually expressed by an Special Trinomial
equation of the form f(x) 5 “an algebraic expression
A trinomial that comes from a special identity.
in x,” is used to calculate the value of the dependent
variable corresponding to each value of the Examples:
independent variable.
1) Square of a sum: (a 1 b)2 5 a2 1 2ab 1 b2
2) Square of a difference: (a 2 b)2 5 a2 2 2ab 1 b2

142 GLOSSARY
 TABLE OF
CONTENTS

Square of a Binomial Symmetric Form

The square of a binomial is equal to the square of The writing of an equation of a straight line in
the form _​​  a ​ 1 _​  b ​5 1​
x y
the first term of the binomial plus the double of the
product of the two terms plus the square of the
Parameter a corresponds to the x-intercept of the
second term.
straight line and parameter b is the initial value.
Example:
(a 1 b)2 5 a2 1 2ab 1 b2 System of Equations
A set of equations that can be tested simultaneously.
Square of Double the Square of
the 1st product of the 2nd
term the 2 terms term Example:
Below is a system of two-variable first-degree equations.
Standard Form
The writing of an equation or a function rule in a { y 5 3x 2 1
​​ ​
y5 x15
​ ​​​

form that helps to identify certain parameters, The solution to the system is the pair (3, 8), which are the
thereby making it easier to graph. only values of x and y that work with both equations.

Example:
Tangent Line to a Circle
The standard form of a first-degree polynomial
A straight line that touches a circle at a single point.
function is f(x) 5 ax 1 b.
The tangent line is always perpendicular to a radius
of the circle.
Step Function
Example:
A function whose domain can be separated into
intervals such that, when the function is limited to Radius
each interval, it remains constant. On a graph, a step
function is represented by horizontal line segments.
Tangent line
Example:
Graph of a step function.

y
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10 Term
9
8 In an algebraic expression, the terms are the
7 different parts of the expression separated by the
6
5 operation symbols 1 and 2.
4
3 Examples:
2
1) 2xy is a term resulting from the multiplication of the
glossary

1
number 2 and its variables x and y.
0 1 2 3 4 5 6 7 8 9 10 x
2) The polynomial 3x2 1 2x −10 is made up of three
terms: 3x2, 2x and 10.

143
 TABLE OF
CONTENTS

Translation Zero (of a function)

A geometrical transformation whereby all the points The value of the domain whose image under the
of a figure are moved in the same direction and by function is equal to 0. The zeros of a function
the same distance. All the characteristics of the correspond to the x-intercepts on its graph.
translation can be described by a translation arrow.
Example:
Example:
y

Aˈ 7
A 6
5
4
Bˈ 3 Zero
Cˈ 2 of a function
t Translation arrow 1
B C
22 1
2 1 2 3 4 5 6 7 x
2
2

Trinomial
A polynomial with three terms.

Unknown
This book is the property of Dickson Joseph.

In a math problem, an unknown is a value that you

want to determine using one or more equations.

Vanishing Point
A point used to represent a three-dimensional
object in perspective.
Line segments that are parallel in reality are not
necessarily so in a perspective drawing. The
extensions of these segments, called “vanishing
lines,” converge at the vanishing point.

Example:

© SOFAD / All Rights Reserved.

Vanishing point Vanishing point

Vanishing lines Vanishing lines

Vertex (of a parabola)

The point on a parabola located at its axis of
symmetry. For a second-degree polynomial
function, the vertex of the parabola that represents
it graphically can be used to determine the
maximum or minimum of the function.

144 GLOSSARY
 TABLE OF
CONTENTS

ANSWER KEY
CHAPTER 1
SITUATION 1.1
PAY-AS-YOU-GO CALLS
EXPLORATION 1.1 PAGES 5 TO 6
1 In the first contract, the cost is $0.40 for every minute whether 5 a) Independent variable: length of a call (s).
that minute is complete or not. In the second contract, the
b) Dependent variable: cost of a call ($).
cost is $0.20 for every complete 30-second block.

ANSWER KEY
CHAPTER 1
6 Cost of a call as a
2 Contract 1: function of its length
Cost
To calculate the cost of a call based on its length (s), ($)
proceed as follows: 2.00
1.80
Divide the number of seconds by 60 to express the 1.60
length (min). 1.40
1.20
For 100 s: 100 4 60  1.67. 1.00
0.80
Then round the number of minutes up to the nearest integer.
0.60
1.67 is rounded to 2. 0.40
0.20
0.40 3 2 5 0.80. The cost of a 100-s call is $0.80.
0 60 120 180 240 300
Contract 2: Length of call
(s)
To calculate the cost of a call based on its length (s),
proceed as follows:
Divide the number of seconds by 30 to find out how many ACQUISITION 1.1A PAGES 7 TO 15
30-s blocks the call lasted for. 1 a) ]28, 24]; ]24, 4]; ]4, 5] and ]5, 7]
For 100 s: 100 4 30  3.33.
b) Domain: ]28, 7] Range: {23, 0, 2, 5}
Then round the number of 30-s blocks down to the
nearest integer, which represents the number of complete 2 a) 1) f(10) 5 1 2) f(20) 5 1 3) f(20.5) 5 2 4) f(60) 5 3
blocks only.
3.33 is rounded to 3. EXPLANATION: For example, to understand that
f(10) 5 1, you have to look at the graph to see the value of
0.20 3 3 5 0.60. The cost of a 100-s call is $0.60.
the y-coordinate when the value of the x-coordinate is
10 min. In this case, if x 5 10 min, then y 5 $1.00.
3 a) Contract 1:
For 200 s: 200 4 60  3.33. b) The closed circles indicate that the points belong to the
© SOFAD / All Rights Reserved.

3.33 min is rounded to 4 min. function, while the open circles indicate that the points
are excluded.
0.40 3 4 5 1.60. The cost of a 200-s call is $1.60.
Contract 2: 3 a) All values greater than 0 g, up to a maximum of 500 g
For 200 s: 200 4 30  6.67.
b) The possible costs are $6, $8 or $12.
6.67 is rounded to 6 blocks of 30 s each.
c) The critical values are 200 and 300.
0.20 3 6 5 1.20. The cost of a 200-s call is $1.20.
d) 1) 299 g 5 $8.00
b) Contract 1:
2) 300 g 5 $8.00
For 300 s: 300 4 60 5 5. 3) 301 g 5 $12.00
0.40 3 55 2.00. The cost of a 300-s call is $2.00.
e) It weighs between 0 g (exclusive) and 200 g (inclusive).
Contract 2:
For 300 s: 300 4 30 5 10. 4 a) The critical values are {22, 0, 2, 4, 6}.
0.20 3 10 5 2.00. The cost of a 300-s call is $2.00. The range is {26, 23, 0, 3, 6, 9}.

b) The critical values are {24, 0, 4, 8, 12}.

4 30 s The range is {21, 20.5, 0, 0.5, 1, 1.5}.

145
 TABLE OF
CONTENTS

5 The integer part symbol represents a number rounded off to c) x h(x) 5 30.5x
the greatest integer less than or equal to that number. [24, 22[ 26

[22, 0[ 23

6 a) [12] 5 12 [0, 2[ 0
[2, 4[ 3
b) [5.99999] 5 5
[4, 6[ 6
c) ​​[__
​   ​]​​ 5 0
1 [6, 8[ 9
2
d) [23.4 1 1.2] 5 [22.2] 5 23 y
*Caution: the integer below 22.2 is 23, not 22.
8

23
22.222 2 1 0 1 2 3 R 6

4
NOTE: Graphic calculators have an integer part function, so
you can use one to check your answers to question 6. 2

7 You first need to consider the graph for the function f(x) in 24 2 2 2 4 6 8 x
order to correctly interpret the parameters in a rule of the 2 2
form f(x) 5 abx. Then observe the length of a step and the 2 4
height of a jump.
2 6
For g(x): g(x) 5 2x
a 5 1 and b 5 2 11 The steps have a length of 0.5 and the jumps have a height
This book is the property of Dickson Joseph.

1 of 1, so a 5 1 and b 5 2. Therefore, the rule is f(x) 5 2x],

Length of one step: __
​​   ​​
2 where x represents the quantity of lavender (kg) and f(x),
Height of one jump: 1 the quantity of oil produced (ml).
For h(x) 5 30.5x a 5 3 and b 5 0.5
Length of one step: 2 12 a) f(4) 5
 50.25 3 4] 5 51] 5 5 3 1 5 5

Height of one jump: 3 b) f(26) 5

 50.25 3 26] 5 521.5] 5 5 3 22 5 210

c) 20 5 50.25x]
8 a) Height between two steps: 2 20
___
​​   ​​ 5 0.25x]
b) Length of one step: 4 5
4 5 0.25x]
c) Function rule: f(x) 5 20.25x] Since the integer part of 0.25x must be 4, this means
that the result of 0.25x may produce all the values
EXPLANATION: Parameter b is linked to the length of the between 4 (inclusive) and 5 (exclusive). Thus,
step. Here the length of the step is 4, while parameter b is 0.25. 4  0.25x , 5
1
You may have guessed that b 5 ______________
​​   ​​. 16  x , 20
Length of step

© SOFAD / All Rights Reserved.

Therefore, x  16, 20
1
9 For f(x): a 5 0.5 and b 5 __
​​   ​​ 
4 d) 240 5 50.25x] e) 6 5 50.25x]
Length of one step: 4 ___240
__ 6
​​   ​​ 5 0.25x] ​​   ​​ 5 0.25x]
Height of one jump: 0.5 5 5
28 5 0.25x] 1.2 5 0.25x]
For g(x): a 5 2 and b 5 2 28  0.25x , 27
No solution, because it is
Length of one step: 0.5 232  x , 228 impossible for an integer
Height of one jump: 2 Therefore, part to be equal to 1.2.
x  232, 228
10 a) a 5 3 and b 5 0.5
13 a) 1) f(4) 5 4[0.5 3 4] 5 4[2] 5 4 3 2 5 8
b) The critical values are {…, 24, 22, 0, 2, 4, 6, …}.
2) f(27) 5 4[0.5 3 27] 5 4[23.5] 5 4 3 24 5 216

b) 1) 4 5 4[0.5x] 2) 6 5 4[0.5x]
1 5 [0.5x] 1.5 5 [0.5x]
1  0.5x , 2 No solution, because it is
2  x , 4 impossible for an integer
part to be equal to 1.5.

146 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

SOLUTION
Pay-as-You-Go Calls PAGES 16 TO 17
Representation of the Situation Function rule
1
Sample representation (table of values or graph): The steps are 30 units in length, so b 5 ___
​​   ​​.
30
Cost of a call as a function of its length The jumps are 0.20 unit in height, so a 5 0.2.
Length (s)
[0, 30[
Cost of call ($)
0
The rule is therefore C(x) 5 0.2​​ ___ [ ]
1
​   ​  x ,​​ where C(x)is the cost
30
[30, 60[ 0.20 of a call ($) and x, its length (s).
[60, 90[ 0.40 Solution

ANSWER KEY
CHAPTER 1
[90, 120[ 0.60 Sample justification:
[120, 150[ 0.80
Since Annie’s monthly budget is $30.00 and she is likely to
[150, 180[ 1.00
make about 10 calls a month, the average budget per call is
[180, 210[ 1.20
30.00 4 10 5 $3.00.
[210, 240[ 1.40
[240, 270[ 1.60 Length of a call costing $3.00:

Cost of a call as a [ ]
3 5 0.2​​ ___
1
​   ​  x ​​
30

[ ]
function of its length 1
Cost 15 5 ​​ ___
​   ​  x ​​
($) 30
2.00 1
15  ​​ ___ ​​  x , 16
1.80 30
1.60 450  x , 480
1.40
Thus, the average length of a call should be within the interval
1.20
450, 480 s. This corresponds to 7 min 30 s (inclusive) up to
1.00
8 min (exclusive).
0.80
0.60
0.40
0.20

0 30 60 90 120 150 180 210 240 270 300

Length
(s)

d) The function is decreasing.

ACQUISITION 1.1B PAGES 18 TO 21
1 a) 1) f1(23) 5 230.5 3 23] 5 2321.5] 5 23 3 22 5 6 2 a) Sample answers:
2) f1(22) 5 230.5 322] 5 2321] 5 23 3 21 5 3 f2(22) 5 320.5 3 22] 5 31] 5 3 3 1 5 3
© SOFAD / All Rights Reserved.

3) f1(21) 5 230.5 3 21] 5 2320.5] 5 23 3 21 5 3

f2(21) 5 320.5 3 21] 5 30.5] 5 3 3 0 5 0
4) f1(0) 5 230.5 3 0] 5 230] 5 23 3 0 5 0
f2(0) 5 320.5 3 0] 5 30] 5 3 3 0 5 0
5) f1(1) 5 230.5 3 1] 5 230.5] 5 23 3 0 5 0
6) f1(2) 5 230.5 3 2] 5 231] 5 23 3 1 5 23 f2(1) 5 320.5 3 1] 5 320.5] 5 3 3 21 5 23
f2(2) 5 320.5 3 2] 5 321] 5 3 3 21 5 23
b) y
b) y
8
6 8
4 6
2 4
2
28 26 24 2 2 2 4 6 8 x
4
2 28 6
2 4
2 2 2 2 4 6 8 x
6
2 2 4
8
2 2 6
2 8
c) A reflection over the x-axis

147
 TABLE OF
CONTENTS

c) A reflection over the y-axis CONSOLIDATION 1.1 PAGES 22 TO 25

d) The function is decreasing. 1 a) 1) f(8) 5 6[0.25 3 8] 5 6[2] 5 6 3 2 5 12
2) f(23) 5 6[0.25 3 23] 5 6[20.75] 5 6 3 21 5 26
3 a) y
3) f(5.8) 5 6[0.25 3 5.8] 5 6[1.45] 5 6 3 1 5 6
8 4) f(p) 5 6[0.25 3 p]  6[0.785] 5 6 3 0 5 0
6
b) The range of f is { …, 212, 26, 0, 6, 12, … }.
4
These are all integers and multiples of 6.
2
c) The critical values of f are all integers and multiples of 4.
28 2 6 2 4 2
2 2 4 6 8 x
d) y
4
2

6
2

8
2 10

6
b) 1) f3(1) 5 2320.5 3 1] 5 2320.5] 5 23 3 21 5 3 2
2) f3(21) 5 2320.5 3 21] 5 230.5] 5 23 3 0 5 0
210 2 6 2
2
2 6 10 x
3) f3(2) 5 2320.5 3 2] 5 2321] 5 23 3 21 5 3
6
2
4) f3(22) 5 2320.5 3 22] 5 231] 5 23 3 1 5 23
10
2

4 a) y

2 a) 1) f(212) 5 210[20.1 3 212] 5 210[1.2] 5 210 3 1 5 210

This book is the property of Dickson Joseph.

4
2) f(26) 5 210[20.1 3 26] 5 210[22.6] 5 210 3 23 5 30
2
b) It is increasing, because parameters a and b both have
2 2 1
2 1 2 x the same sign.
22

4
2
c) 1) The direction of the steps is reversed—they go from
right to left ( )—because b is negative.
2) The length of the steps is 10 units because b 5 20.1
b) y and so ____
1
​​  2  ​​ 5 ___
1
​​  ​​ 5 10.
​| 0.1|​ 0.1
8 3) The height of the jumps is also 10 units because
6 |a| 5 10.
4 d) y
2
20
15
2 8 2 6 4
2 2 2 2 4 6 8 x 10
2 2
5

© SOFAD / All Rights Reserved.

20 215210 25 5 10 15 20 x
5 The rules are f(x) 5 222x] and g(x) 5 2324x].
2

210

215

220
EXPLANATION:
For the function f:
3 a) Plane 1
Since the closed endpoints are on the right side of the steps,
y
parameter b is negative; since the function is decreasing,
parameter a has the opposite sign, so it is positive. Since the 6
height of the jumps is 2, the value of parameter a is 2; since the C D
1 4
length of the steps is 0.5, the value of parameter b is ___
​​   ​​ 5 2.
0.5 A B
For the function g: 2

Since the closed endpoints are on the right side of the steps,
parameter b is negative; since the function is increasing, 26 4
2 22 2 4 6 x
parameter a has the same sign, so it is also negative. Since the 2
2

height of the jumps is 3, the value of parameter a is 23; since Aˈ Bˈ

4
2
the length of the steps is 0.25, the value of parameter b is
1
2​____
Cˈ Dˈ
​  ​​ 5 24. 6
2
0.25

148 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

Plane 2 b) a 5 20   b 5 20.2

y g(x) 5 20[20.2x]
6
Dˈ Cˈ C D EXPLANATION: The intervals for which the function is
4 constant are open on the left. For example, in the interval
Bˈ Aˈ A B
]25, 0], the lower endpoint of 25 is not included. This means
2
the steps in the graph are oriented like this: . The
value of b is therefore negative. The length of the intervals,
26 4
2 2 2 2 4 6 x which is equal to 5, matches the length of the steps. As a
2
2 1
result, b is equal to 2​__
​ ​​, or 20.2.
5
4 The function is decreasing: when the value of x increases in

ANSWER KEY
2

CHAPTER 1
the table, g(x) is stable or decreases. Since the function is
6
2
decreasing, a and b do not have the same sign; therefore, a is
positive. Since the value of the dependent variable changes by
Plane 3 20 units from one interval to the next, the height of the jumps
y is 20. The value of a is therefore 20.

6
C D 7 a) 1) Since there are 3 full blocks of 30 min in 1 h 40 min,
4 the cost will be 3 3 1.50 5 $4.50.
A B
2 2) Since there are 4 full blocks of 30 min in 2 h, the cost
will be 4 3 1.50 5 $6.00.
2 6 4
2 2 2 2 4 6 x b)
2
2 Cost as a function
Bˈ Aˈ of length of stay
Cost
4
2
($)
Dˈ Cˈ 12
6
2

10
b) Plane 1: a negative parameter a
8
Plane 2: a negative parameter b
Plane 3: negative parameters a and b 6

4
4 The equation has no solution. If you attempt to solve it,
you get the equation 1.5 5 0.3x], but it is impossible for 2

an integer part to be equal to 1.5.

0 1 2 3 4
5 a) x  13, 14 Length of stay
(h)
Calculation: 2x] 5 26
c) C(x) 5 1.5[2x], where x is the length of stay (h) and C(x),
x] 5 13
the cost ($).
13  x , 14
© SOFAD / All Rights Reserved.

d) Range of C: {0, 1.5, 3, 4.5, 6, 7.5, 9, 10.5, 12}

b) x  250, 240
Calculation: 25 5 250.1x]
8 a) 1) 2nd lap 2) 3rd lap 3) 3rd lap
25 5 0.1x]

25  0.1x , 24 b) He has run more than 1200 m (the equivalent of 3 laps)

250  x , 240 but has not passed 1600 m (otherwise, he would be on
his 5th lap). The distance covered (m) by the runner is
6 a) 
a 5 22  b 5 0.5 therefore within the interval ]1200, 1600].
f(x) 5 22[0.5x] c) Table of values
Distance covered (m) Lap number
EXPLANATION: According to the direction of the ]0, 400] 1
steps—from left to right ( )—the value of b is positive. ]400, 800] 2
Since the length of the steps is 2 units, you can conclude that
1 ]800, 1200] 3
b is equal to __
​​  ,​​ or 0.5.
2 ]1200, 1600] 4
Since the function is decreasing, a and b do not have the same ]1600, 2000] 5
sign; therefore, a is negative. Since the height of the jumps is
2 units, the value of a is 22.

149
 TABLE OF
CONTENTS

Graph [
L(x) 5 2​​ 2___
x
]
​400 ​ ​​, where x represents the distance covered (m)
and L(x), the number of the lap the runner is currently on.
Lap number as a function
of distance covered
Lap
number NOTE: The rule may be written in two other ways:

[ ]
5 1 ​
L(x) 5 2​​ 2____
​   ​  x ​or L(x) 5 220.0025x
4
400

[ ]
3
600
e) 1) L(600) 5 2​​ 2​ ____ ​ ​​ 5 221.5 5 2(22) 5 2
2 400

1 2) L(900) 5 ​ ​ 
2​ 2
[ ]
900 ​ 2 2
____
400
 ​ ​5  2.25 5 2(23) 5 3

0 400 800 1200 1600 2000

[ ]
3) L(1200) 5 2​​ 2_____
​ 
1200 ​ 2 2
400
 ​ ​5  3 5 2(23) 5 3
Distance f ) dom L 5 ]0, 2000]
covered
(m) ran L 5 {1, 2, 3, 4, 5}

d) According to the direction of the steps, b is negative. NOTE: The number 0 is not part of the domain because the
1
The steps are 400 units in length. Therefore, |b| 5 ___
​​  .​​ runner cannot be said to be running any particular lap when
1 ​ 400
___
Consequently, b is equal to ​ ,​ or 0.0025.
2​ 2 he is on the starting line. He must start running to begin his
400
first lap.
Since the function is increasing, a and b have the same
sign; therefore, a is also negative. Since the height of the
jumps is 1 unit, the value of a is 21.
This book is the property of Dickson Joseph.

SITUATION 1.2
MEASURING THE TEMPERATURE
EXPLORATION 1.2 PAGES 27 TO 28
1 a) Type of rounding 2.05 °C 2.50 °C 2.65 °C 3 In the graph of the greatest integer function f(x) 5 [x], the
1) Up to the nearest integer 3 °C 3 °C 3 °C critical values are integers, while in this case, they are values
2) Down to the nearest integer 2 °C 2 °C 2 °C halfway between two integers: 0.5, 1.5, 2.5, etc.
3) To the nearest integer 2 °C 3 °C 3 °C
ACQUISITION 1.2A PAGES 29 TO 37
b) Rounding to the nearest integer provides more accurate
1 a) This graph represents the greatest integer function
information than simply rounding up or down. As you
f(x) 5 x], where x represents the age (years) and f(x),
can see in the table for question 1 a), the difference
the dose (ml).
between the number and its rounded value is never
more than 0.5. The difference can be greater with other b) The first step at 0 reflects the fact that children under
types of rounding; for example, if rounding down, the the age of 1 should not take this medication.

© SOFAD / All Rights Reserved.

difference between 2.65 °C and its rounded value will
2 a) Dose of medication
be 0.65. based on child’s age
Dose
(ml)
2 6
Temperature rounded to nearest integer
5
based on real temperature
Rounded 4
temperature
(°C) 3 3
2
1
2
0 1 2 3 4 5
Age
1 (y)

b) Since the jumps now have a height of 2, a 5 2 and the

0 1 2 3 rule is f1(x) 5 2x], where x represents the age (years)
Real temperature and f1(x), the dose (ml).
(°C)
c) A vertical stretch away from the x-axis.

150 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

3 a) Dose of medication 6 The graph has undergone the following transformations.

based on child’s age
Dose 1) Since a is negative, the graph has undergone a
(ml)
6 reflection over the x-axis.
5 2) Since the value of |a| is 2, it has undergone a vertical
4 stretch of 2.
3
3) Since the value of b is 5, it has undergone a horizontal
2
compression of 5.
1
4) Since the value of h is 21, it has undergone a horizontal
0 1 2 3 4 5 translation of 1 unit to the left.

ANSWER KEY
CHAPTER 1
Age
(y) 5) Since the value of k is 24, it has undergone a vertical
translation of 4 units down.
b) Since the length of the steps is now 0.5, f2(x) 5 22x],
where x represents the age (years) and f2(x), the dose (ml). 7 a) (16, 210)
c) A horizontal compression toward the y-axis. b) The steps are 8 units in length.

c) Since parameter b is positive, the closed endpoints are

4 a) Dose of medication
based on child’s age on the left.
Dose
(ml) d) Since parameters a and b have opposite signs, the
5
function is decreasing.
4
3 e) The height of the jumps is 10 units.
2
f) y
1
10
8
0 1 2 3 4 5
6
Age
(y) 4
2
b) Since 1 ml is added at the end of the calculation,
f3(x) 5 22x] 1 1, where x represents the age (years)
22 2 4 6 8 10 12 14 16 18 20 22 24 x
24
and f3(x), the dose (ml). 26

28
c) A vertical translation of 1 unit up.
210

5 a) Dose of medication
1
based on child’s age b 5 __
8 a)  ​​  ​​because the steps are 2 units long and go from a
Dose 2
(ml) closed endpoint to an open endpoint; a 5 21.5 because
5 the jumps have a height of 1.5 and, since the function is
© SOFAD / All Rights Reserved.

4 decreasing, a and b must have opposite signs.

3
b) Sample answer: h 5 1 and k 5 4
2
1
EXPLANATION: Since the graph has an infinite number of
closed endpoints, many pairs (h, k) could serve as parameters
0 1 2 3 4 5 of the function. By convention, the closed endpoint closest
Age
(y)
to the origin, and usually in the first quadrant, is used to
write the rule.
b) Since 0.5 year has been added to the age,
f4(x) 5 22(x 1 0.5)] 1 1, where x represents
the age (years) and f4(x), the dose (ml).
c) A possible rule for the function is f(x) 5 21.5 ​​ __
1
[ ]
​   ​  (​x 2 1) ​​ 1 4.
2

c) A horizontal translation of 0.5 unit to the left 9 A possible rule is f(x) 5 22[2(x − 1)] 1 2.5,
where x represents the number of days late
and f(x), the amount of the late fee.
The amount of the late fee:
f(2) 5 22[2(2 − 1)] 1 2.5 5 22[21] 1 2.5 5 2 1 2.5 5 4.5
5 $4.50

151
 TABLE OF
CONTENTS

10 a) When the fractional part reaches five tenths, b) You have probably noticed that, in the standard form of
or “point-five.” the rule, f(x) 5 ab(x − h) 1 k, parameter h is subtracted
from the independent variable. Therefore, when parameter
b) Yes, for positive and negative integers alike.
h is negative, you need to subtract a negative number—or,
y more simply, add the opposite (positive )number.
11
Nearest You thus get f(x) 5 x − (20.5) 5 [x 1 0.5].
3 integer
function In the graph for question 11, the two rules can be
compared as follows: the jumps in the nearest integer
function begin 0.5 units before those in the greatest
2
integer function.

13 1) 23.4 becomes 23.

1 2) 23.5 becomes 24.
3) 23.6 becomes 24.
f(x)
14 1) f(23.4) 5 [23.4 1 0.5] 5 [22.9] 5 23
0 1 2 3 x 2) f(23.5) 5 [23.5 1 0.5] 5 [23] 5 23
*Rounding to the nearest integer
12 a) The function does seem to round positive numbers to
would result in a value of 24.
the nearest integer. Sample checks:
3) f(23.6) 5 [23.6 1 0.5] 5 [23.1] 5 24
f(1.5) 5 [1.5 1 0.5] 5 [2] 5 2
f(3.95) 5 [3.95 1 0.5] 5 [4.45] 5 4
f(3.2) 5 [3.2 1 0.5] 5 [3.7] 5 3
This book is the property of Dickson Joseph.

SOLUTION
Measuring the Temperature PAGES 38 TO 39
The Function for Rounding Off the Temperature
EXPLANATION: For negative temperatures, when the
Sample representation (table of values or graph): number in the tenths position is 0, 1, 2, 3 or 4, it is rounded
up to the integer above, and when the number in the tenths
a) Possible rule: f(x) 5 [x 1 0.5]  if x  [0, 1∞ °C position is 5, 6, 7, 8 or 9, it is rounded down to the integer
below. The critical values are thus still halfway between two
where f(x) is the temperature rounded to the nearest integer
integers, but the closed endpoints are on the right side of
for a real temperature of x °C.
the steps.

EXPLANATION: This is the rule established in question 12 c) Possible rule : f(x) 5 2[2(x 1 0.5)] 2 1  if x  2∞, 0 °C
of the preceding acquisition activity.
where f(x) is the temperature rounded to the nearest integer
b) Temperature rounded to nearest integer for a real temperature of x °C.

© SOFAD / All Rights Reserved.

as a function of real temperature
Temperature rounded to EXPLANATION: It is possible to determine the rule for both
nearest integer (°C)
the positive and negative parts of the domain from the graph.
23 22.5 2
2 2 1.5 1
2 0.5
2 0.5 1 • The steps are 1 unit in length and height in both cases.
20.5
Real
temperature • Parameters a and b have the same sign because the
1
2 (°C) function is increasing.
• According to the direction of the steps, for the positive
1.5
2
part of the domain, b . 0, so both parameters a and b are
2
2 positive (a 5 b 5 1). For the negative part of the domain,
b , 0, so the parameters are negative: a 5 b 5 21.
2.5
2
The values of parameters h and k can be determined using
3
2 the coordinates of an endpoint of a step in each of the parts,
for example, the point (20.5, 21) in the negative part and the
point (0.5, 1) in the positive part.

152 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

{
2[2(x 1 0.5)] 2 1 if x  ]2∞, 0[
d) Possible rule: f(x) 5 e) Temperature rounded to nearest integer
[x 2 0.5] 1 1 if x  [0, 1∞[ as a function of real temperature

{
2[x 1 0.5] if x  [2∞, 0[ Temperature rounded to
Other possible rule: f(x) 5 nearest integer (°C)
[x 1 0.5] if x  [0, 1∞[ 3

This rule is determined by using the point (0.5, 0) for the 2

negative part and the point (20.5, 0) for the positive part.
1

NOTE: This is called a piecewise function because it presents two

distinct rules in its domain. Although this function is not strictly 23 22 21 1 2 3
part of your program, it is useful to know how it is written. 1
2
Real
temperature

ANSWER KEY
CHAPTER 1
2
2 (°C)

3
2

The interval
The real temperature in Mont-Tremblant is within the interval ]20.5, 0.5[ °C when the temperature is reported as 0 °C.
Justification: This answer is based on the set of x-intercepts from the graph. You can see that this set is an open interval
that does not include the endpoints 20.5 °C and 0.5 °C.

b) Sample solutions for determining the number of minutes

ACQUISITION 1.2B PAGES 40 TO 44
that Jade can talk to her aunt for:
1 a) • The intervals are closed on the left side (closed A card with $2.53 on it means that f(x) 5 2.53.
endpoint) and open on the right (open endpoint);
f(x) 5 20.055[2x 1 1] 1 0.055
therefore, b is positive. As seniority increases, so does
2.53 5 20.055[2x 1 1] 1 0.055
the salary. The function is therefore increasing, so
2.475 5 20.055[2x 1 1]
parameter a is also positive. 245 5 [2x 1 1]
• The salary (dependent variable) increases by $3000 at 245  2x 1 1  244
a time, so each jump has a height of 3 and a 5 3. 246  2x  245
• The difference between the beginning and end of
46  x  45 or 45  x  46
each interval is 2 (6 − 4 5 2), so each step has a length
1
of 2 and b 5 __
​​  .​​
2 NOTE: When an inequality is multiplied or divided by a
• One of the closed endpoints is at (0, 35), so possible
negative number, the “greater than” and “less than” signs
values for parameters h and k are h 5 0 and k 5 35. are reversed.
[ ]
b) Sample possible rule: f(x) 5 3​​ ​ _2 ​  x ​​ 1 35, where x represents
1
With $2.53 on her card, Jade could talk to her aunt for
seniority and f(x), the salary in thousands of dollars.
more than 45 min, up to a maximum of 46 min.
2 a) Sample solution for determining the rule:
c) A call of 26.5 min means that x 5 26.5.
1. One of the closed endpoints is at (1, 0.055),
© SOFAD / All Rights Reserved.

so possible values for parameters h and k are h 5 1 f(x) 5 20.055[2x 1 1] 1 0.055

and k 5 0.055. f(26.5) 5 20.055[226.5 1 1] 1 0.055
2. The cost increases by $0.055 per min, so the height of 5 20.055[225.5] 1 0.055 5 20.055(226) 1 0.055
a jump is 0.055 and |a| 5 0.055. 5 1.43 1 0.055 5 1.485
3. Each interval lasts 1 min, so the length of each step The call would cost $1.49.
is 1 and |b| 5 1.
4. The intervals are closed on the right side (closed 3 a) For a period of ]0, 14]. This is the domain of the function.
endpoint), so b is negative. b) The amounts are {70, 120, 170, 220, 270, 320, 370}.
5. As the length of the call increases, so does its cost. This is the range of the function.
The function is therefore increasing, so parameter a
c) The minimum amount is $70.
is also negative.
This is the minimum of the function.
Sample possible rules:
f(x) 5 20.055[−(x 2 1)] 1 0.055 or d) The maximum amount is $370.
f(x) 5 20.055[2x 1 1] 1 0.055, This is the maximum of the function.

where x represents the length of the call (min) e) As the number of days increases, so does the cost.
and f(x), the cost ($). The function is therefore increasing.

153
 TABLE OF
CONTENTS

4 a) Domain: set of real numbers 2 a) The right-hand expression in the rule can be transformed
by the following algebraic manipulations:
b) Codomain: {…, 25, 23, 21, 1, 3, …}
4 − 2[3 1 0.5x] 5 22[3 1 0.5x] 1 4 (commutative property
c) x-intercept: none
of addition)
d) y-intercept: 1 5 22[0.5x 1 3] 1 4 (commutative property
e) Sign: The function is positive over x  ]2∞, 0]. of addition)
The function is negative over x  ]0, 1∞[. 5 22[0.5(x 1 6)] 1 4 (factoring out the
f ) Direction: The function is decreasing over the entire domain. coefficient of x)
Therefore, f(x) 5 22[0.5(x 1 6)] 1 4.
g) Extrema: none
b) a 5 22; b 5 0.5; h 5 26; k 5 4
h) Function rule: f(x) 5 2[2x] 1 1
c) y
5 a) Funds raised as a
function of time 6
Amount
($)
6 4

5 2

4
2 6 2 4 22 2 4 6 x
3 2 2
2
2 4
1
This book is the property of Dickson Joseph.

6
2

0 20 40 60 80 100
d) The range of f is {…, 24, 22, 0, 2, 4, …}.
Time
(s)
e) The function is decreasing.

b) 1) Domain: 0, 60] f ) The zero set is the interval [22, 0[.
2) Codomain: {0, 1, 2, 3, 4, 5, 6}
1
3 a) a 5 4, b 5 __
​​   ​​, h 5 1 and k 5 2
3) x-intercepts: 0, 10 3
4) y-intercept: 0 [
f (x) 5 4​​ __
1
​   ​  (​x 2 1) ​​ 1 2
3 ]
5) Sign: The function is positive over the entire domain.
6) Direction: The function is increasing over the entire NOTE: There are several possible values for parameters h
and k. For example, their values could also be h 5 22 and
domain.
7) Extrema: minimum of 0 and maximum of 6
k 5 22. In this case, the rule would be f(x) 5 4 ​​ __
1
[3 ]
​  ​  (​x 1 2) ​​ − 2.

8) Function rule: f(x) 5 ​​ ___[ ]

x
​   ​ ​​
10 b) a 5 2.5, b 5 20.25, h 5 21 and k 5 5

© SOFAD / All Rights Reserved.

CONSOLIDATION 1.2 PAGES 45 TO 49 g(x) 5 2.5[20.25(x 1 1)] 1 5
Once again, several values could be applied to
1 a) The function is decreasing, because a and b have
parameters h and k.
opposite signs.
Another possible equation, with h 5 3 and k 5 2.5,
b) because b is positive. is g(x) 5 2.5[20.25(x − 3)] 1 2.5.
c) 3 units
4 a) y
d) 1 unit
5
e) y 4
3
3
2
2
1
1

x
2 5 24 23 22 21 1 2 3 4 5 x
25 24 23 22 21 1 2 3 4 5
22
22
23
23
24

25

154 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

b) Since a 5 2, the height of the jumps is doubled and the Sample solution:
graph undergoes a vertical stretch of 2 units, going from 1) Since the closed endpoints are on the left side of the
1 to 2. Next, since h 5 23, the function undergoes a steps, parameter b will be positive.
horizontal translation of 3 units to the left. Finally, since
2) Since the function is increasing, parameters a and b
k 5 21, the steps undergo a vertical translation of 1 unit
have the same sign. Therefore, a will be positive.
down.
3) Since the length of the steps is 5 units, the value of
1
5 A possible equation for the greatest integer function parameter b will be __
​​  ​​ or 0.2.
5
[5 ]
represented by the table of values is f(x) 5 2​​ __
1
​  ​  x ​1 6. 4) Since the height of the jumps is 8 units, the value of
Once again, since several values could be applied to parameter a will be 8.
parameters h and k, other equations are possible, such as 5) For the pair (h, k), you can use (12, 0). Another

ANSWER KEY
CHAPTER 1
g(x) 5 2[0.2(x 1 5)] 1 4, when h 5 25 and k 5 4. possibility is (17, 8).

Sample solution: b) Sample solution:

1) Since the brackets around the the intervals are closed on This value can be determined by extending the sketch of
the left and open on the right, the endpoints go from the graph (downwards), following the pattern.
closed to open. Therefore, b will be positive.
It is also possible to use the rule. Finding the y-intercept
2) Since the function is increasing, parameters a and b have means finding the value of y when x 5 0—in other
the same sign. Therefore, a will be positive. words, calculating f(0).

3) Since the length of the steps is 5 units, the value of f(0) 5

 8[0.2(0 − 12)] 5 8[0.2(212)] 5 8[22.4] 5 8 3 2 3 5 224
1
parameter b will be __
​​  .​​ The y-intercept is 224.
5
4) Since the height of the jumps is 2 units, the value of
7 A possible equation for the greatest integer function
[8 ]
parameter a will be 2. 1
represented by the graph is f(x) 5 4​​ 2__
​ ​  (​x − 4) ​​ − 2.
5) For the pair (h, k), you can use (0, 6). a) Sample solution:
1) Since the closed endpoints are on the right side of the
6 a) 
Many rules are conceivable depending on the choice of
steps, parameter b will be negative.
values for h and k.
2) Since the function is decreasing, parameters a and b
One of the simplest rules is written as follows:
have opposite signs. Therefore, a will be positive.
f(x) 5 8[0.2(x − 12)].
3) Since the length of the steps is 8 units, the value of
1
EXPLANATION: Based on the information provided, you parameter b will be 2__
​​   ​​.
8
can draw the following sketch. 4) Since the height of the jumps is 4 units, the value of
y parameter a will be 4.
5) Various pairs (h, k) are possible, such as (4, 22) or (24, 2).
(17, 8)
b) Increasing parameter a by 2 units means that the jump is
increased by 2 units (going from 4 to 6 units). The
© SOFAD / All Rights Reserved.

function would undergo a vertical stretch.

12 17 x c) The function would undergo a horizontal translation of

3 units to the left.

8 a) 1) 100  2) 120  3) 240

b) The function rounds a positive number to the nearest 10.

[ ]x15
c) f(x)5 10​​ ____
​ 
10
1
 ​​​or, in standard form, f(x) 5510​​ ___ [ ]
​   ​  (​x 1 5) ​​
10
It is possible to find the answer by filling in the diagram as follows:

x
Add
5
x5 Divide
by 10 [x__
10 ]
5 Apply
integer part 10 ]
[x__
5 Multiply
by 10 [ ]
x5
10 __
10

155
 TABLE OF
CONTENTS

9 a) Number of days since loan Debt ($)

11 a) There are several possible rules.
[0, 7[ 500 [
R(x) 5 250 ​​ 2__1
]
​40 ​  (​x 2 40) ​​ 172.99, where R represents the
[7, 14[ 450 cost of renting the buses and x, the number of people.
[14, 21[ 400
b) If 239 people sign up, Matthew has to pay Locabus $322.99.

[ ]
[21, 28[ 350 1
[28, 35[ 300
R(239) 5 250 ​​ 2___​   ​  (​239 2 40) ​​ 1 72.99
40
[35, 42[
[42, 49[
250
200 [
5 250 ​​ 2___
1
]
​   ​  (199) ​​ 1 72.99 5 250[24.975] 1 72.99
40
[49, 56[ 150 5 250 3 25 1 72.99 5 $322.99
[56, 63[ 100 c) A maximum of 280 people, but more than 240 people
[63, 70[ 50 have signed up.

[ ]
70 0 1
372.99 5 250 ​​ 2​ ___ ​  (​x 2 40) ​​ 1 72.99
40
Oliver’s debt depending on
number of days since loan
300 5 250 ​​ 2___ [
1
]
​   ​  (​x 2 40) ​​
40
Debt
($)
500
[ 1
26 5 ​​ 2___
]
​   ​  (​x 2 40) ​​
40
450
400
350
[
26  2​​ 2___ 1
]
​   ​  (​x 2 40) ​​ 25
40
300 240  x − 40  200
250
200
150 NOTE: When an inequality is multiplied or divided by a
100
50 negative number, the “greater than” and “less than” signs
This book is the property of Dickson Joseph.

are reversed.
0 7 1421 28 35 42 49 56 63 70
280  x  240
Number of days
since loan

b) The simplest rule is D(x) 5 250​​ __

x
[]
​   ​ ​​ 1 500.
7
INTEGRATION PAGES 56 TO 61
c) After 30 days, he will owe his father $300. 1 a) 1) 4 cups
Possible justifications: You can refer to the table of values 2) 29 cups
or the graph, noting that 30 is within the interval [28, 35[. 3) 64 cups
It is also possible to solve D(30) using the rule.
b) Sample solution:
D(30) 5 [ ]30
 250​​ ___ [ ] 2
​   ​ ​​ 1 500 5 250​​ 4 ​ __ ​ ​​ 1 500
7 7 • Multiply the amount of water by 1000 to get the
5 250 3 4 1 500 5 300 number of millilitres.
• Divide this number by 250 to get the number of cups
10 a) The base number of workers, regardless of the number of including any fraction of a cup.
children: there must be at least 2 workers on staff at all
• Round the result up to the nearest integer.
times, even if no children are registered.

© SOFAD / All Rights Reserved.

b) For every group of 7 children, another worker must be c) The simplest rule for the function is N(x) 5 2[24x].
added. Sample solution:
c) If 43 children are registered, 8 workers will be needed. 1) The closed endpoints are on the right side of the
Calculation: W(x) 5 ​​ __ [] x
​   ​ ​​ 1 2
7
steps. For example, 1 cup is needed for more than 0 L
up to 0.25 L (250 ml) inclusive, or the interval ]0, 0.25].
W(43) 5 ​[​ ___​   ​]​1 2 5 6.14 1 2 5 6 1 2 5 8
43
This means that parameter b is negative.
7
d) If 14 workers are employed, a minimum of 84 children 2) The function is increasing (the more water there is in
and a maximum of 90 children will be attending the the cooler, the greater the number of cups), and since
daycare centre. parameters a and b have the same sign, a is negative.
Calculation: W(x) 5 ​​ __ []x
​   ​ ​​ 1 2
7
3) Since the length of the steps is 0.25 unit, the value of
parameter b will be 24.
[]x
14 5 ​​ ​ __ ​ ​​ 1 2
7 4) Since the height of the jumps is 1 unit, the value of
[]x
12 5 ​​ ​ __ ​ ​​
7
parameter a will be 21.

[]x 5) You can use the pair (0, 0) for (h, k) (if there is no water
12  ​​ ​ __ ​ ​​ , 13
7 in the cooler, no cups are needed to empty it).
84  x , 91

156 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

d) 6 a) Number of turns remaining as a

Number of cups to empty a water cooler function of distance covered
as a function of the amount of water Number of turns y
Number remaining
of cups 7
12
6
11
5
10
4
9
8 3
7 2
6 1
5
4 0 50 100 150 200 250 300 350 400 x

ANSWER KEY
CHAPTER 1
Distance
3
(m)
2
1

0 1 2 3
Amount of water
in the cooler
(L) NOTE: The point (400, 0) may be included in the graph,
which would break the pattern but reflect the situation more
e) The cooler contains more than 0.5 L to a maximum of accurately, considering that the domain of the function is the
0.75 L. In other words, the amount of water it contains, interval [0, 400].
in litres, is within the interval ]0.5, 0.75].
b) For all values of the domain that are less than 400,
2 Sample rules for each function: the rule is

a) f(x) 5 3x 1 3 [ ]
x
f(x) 5 2​​ ​ ___ ​ ​ ​1 7.
50
For x 5 400, f(x) 5 0, which may be written as f(400) 5 0.
b) f(x) 5 22​​ __
1
[ ]
​   ​  (​x 2 3) ​​ 1 2 or f(x) 5 22​​ __
4
1
[
​   ​  (​x 2 7) ​​
4 ] c) The zero set is the interval [350, 400].

[ 1
]
c) f(x) 5 22​​ 2 ​ __ ​  (​x 2 1) ​​ − 1 or f(x) 5 22​​ 2__
2 [ 1
]
​   ​  (​x 2 3) ​​ 11
2 d) The set corresponds to the distance covered by the
swimmer when she has no more turns to do. She is
3 The function is decreasing. therefore on the last length of the race.

JUSTIFICATION: The rule can be written in standard form: 7 a) The function is increasing, because the longer the rental
f(x) 5 22[2(2x 2 5)] 1 3. time, the greater the cost.
Since parameters a and b (22 and 2) have opposite signs,
the function is decreasing.
b) The domain is 0, 120 and the range is {65, 105, 145, 185}.

c) Rate
4 The truncation of a positive number is simply the greatest ($)
195
integer function f(x) 5 [x]. For negative numbers, truncation 180
consists in rounding up to the nearest integer, so the rule is
© SOFAD / All Rights Reserved.

165
f(x) 5 2[2x]. These two rules can be combined in a single 150
rule. This therefore makes a piecewise function, and the rule 135
120
can be written as follows:
105

{
2[2x] if x  ]2∞, 0[ 90
f(x) 5
[x] if x  [0, 1∞[ 75
60
5 Sample rules for each function: 45
30

[ 1
]
a) f(x) 5 22​​ 2 ​ __ ​  (​x 2 2) ​​
4
15

9​​[___
​   ​  (​x 2 5)]​​ 1 6
1 0 30 60 90 120
b) f(x) 5 2 Rental time
10
(min)

[2000
x
8 a) S(x) 5 95​​ _____
​  ]
 ​ ​ ​1 250, where S(x) represents the weekly
salary and x, the value of the sales made during the week.

157
 TABLE OF
CONTENTS

b) She needs to make $16 000 worth of sales to earn a 10 Advice for Alexis:
salary above $950. The best choice depends on the download capacity he
Calculation: 950 5 95​​ _____[ ]
​ 
x
2000
 ​ ​​ 1 250 needs, as shown in the table below.
Best choice based on download
[ ]
700 5 95​​ _____ ​ 
x
2000
 ​ ​​ Download (GB) Best choice
[0, 30] Fibre-plus
[ ]
x
7.37  ​​ ​ _____ ​ ​​
2000 ]30, 50] Equivalent packages
Since it is impossible to have an integer part of 7.37, ]50, 1∞[ Cable Distribution

this equation cannot be solved. However, $950 is the Justification:

minimum salary sought. You need to determine the Comparison of suppliers’ costs based on download
amount of sales that would provide a salary of more Download (GB) Cost of Fibre-plus ($) Cost of Cable Distribution ($)
than $950, so 7.37 is rounded off to 8. [0, 10] 20 30

​  [ ]
8  ​​ _____
x
2000
 ​ ​​ , 9
]10, 20]
]20, 30]
25
30
30
35
16 000  x , 18 000 ]30, 40] 35 35
c) If she sells for $36 879, her salary will be $1960. ]40, 50] 40 40
]50, 60] 45 40
​ [ ]
Calculation: S(x) 5 95 ​​ _____
x
2000
 ​ ​​ 1 250
Cost of a package as a
S(36 879) 5 95 ​​[​ ______ ​]​​1 250 5 9518.4395] 1 250
36 879 function of download
2000 Cost
($)
5 95 3 18 1 250 5 1710 1 250 5 1960 50
This book is the property of Dickson Joseph.

40
C(x) 5 28 ​​[2__
9 a)  ​90 ​  (​x 2 90)]​​ 1 14, where C is the cost of
1
30
renting a kayak and x, the number of min.
20
b) The cost of a 4-hour rental is $30.
10
Calculation: 4 h 3 60 min/h 5 240 min

[
C(240) 5 28 ​​ 2___
1
90 ]
​   ​  (​240 2 90) ​​ 1 14
_
0 20 40 60 80 100
Download

[
5 28 ​​ 2___
1
]
​   ​  (150) ​​ 1 14 5 28[21,​​6​​] 1 14
90
Fibre-plus
Cable Distribution
(GB)

5 28(22) 1 14 5 30
The answer can be explained using a graph. To make it
c) The rental time is a value within the interval ]270, 360]
easier to compare costs, the graphs are plotted on the same
min, so over 270 min (4.5 h) up to 360 min (6 h) inclusive.
Cartesian plane.
Calculation:

[ 1
]
38 5 28 ​​ 2​ ___ ​  (​x 2 90) ​​ 1 14
90
EXPLANATION: The graph for Cable Distribution is regular
except for the left endpoint of the first step (if download is
[
24 5 28 ​​ 2___
1
]
​   ​  (​x 2 90) ​​
90
equal to 0). For any other value in the domain, the cost ($)

© SOFAD / All Rights Reserved.

[ ]
with this company may be represented by the function
23 5 ​​ 2___ 1
​   ​  (​x 2 90) ​​ C1(x) 5 25[20.05(x − 20)] 1 30, where x is download in GB.
90
23  2​___ 1 The graph for Fibre-plus is also regular except for the left
​   ​​(x − 90) , 22
90 endpoint of the first step (if download is equal to 0). For any
*When an inequality is multiplied or divided by a other value in the domain, the cost ($) with this company may
negative number, the “less than” and “greater than” be represented by the function C2(x) 5 25[20.1(x − 10)] 1 20,
signs change direction. where x is download in GB.
270  x 2 90 . 180 or 180 , x − 90  270
The graph shows that the cost of the Fibre-plus package is
270 , x  360
lower if the download capacity (in GB) is within the interval
[0, 30]. Furthermore, for downloads within the interval
]30, 40], the two packages are equal: $35.00 for downloads
between 30 (exclusive) and 40 (inclusive) GB and $40.00 for
downloads between 40 (exclusive) and 50 (inclusive) GB.
Finally, if Alexis uses more than 50 GB, the Cable Distribution
package is more economical.

158 CHAPTER 1 – Step Functions and Greatest Integer Functions

 TABLE OF
CONTENTS

LES
Timekeeping
PAGES 62 TO 63
Representation of the Function Modelling The rule: f (x) 5 0.1[10x], where f(x) is the time recorded (s) for a
the Action of the Timer real time of x seconds.

Sample representation:
EXPLANATION: The function essentially consists of rounding down
Real time (s) Time shown on timer (s) to the nearest tenth, regardless of the number in the hundredths
[0, 0.1[ 0 position. For example, 0.86  0.8.

ANSWER KEY
CHAPTER 1
[0.1, 0.2[ 0.1 It is possible to determine the rule from the table of values or the graph.
[0.2, 0.3[ 0.2
1) Since the closed endpoints are on the left side of the steps,
[0.3, 0.4[ 0.3 parameter b is positive.
[0.4, 0.5[ 0.4
2) The function is increasing (as real time advances, so does the
[0.5, 0.6[ 0.5
time on the timer), and since parameters a and b have the same
[0.6, 0.7[ 0.6
sign, a is positive.
[0.7, 0.8[ 0.7
3) Since the length of the steps is 0.1 unit, the value of parameter b
[0.8, 0.9[ 0.8
is 10.
[0.9, 1[ 0.9
[1, 1.1[ 1 4) Since the height of the jumps is 0.1 unit, the value of parameter
a is 0.1.
5) For the pair (h, k), you can use (0, 0).
Time recorded as a
function of real time
Time recorded
(s)
1.2 ANSWER: A recorded time of 16.8 s is compatible with a real time (s)
within the interval [16.8, 16.9[. This interval contains an infinity of
1 numbers.
Therefore, having the same recorded time does not necessarily mean
0.8
that one competitor solved the cube as quickly as the other.
0.6 It is possible, for example, that Waheb Jawad solved the cube in 16.80 s,
while Mike Cheung did it in 16.89 s. In this case, the recorded time
0.4 would be the same for both even though Waheb was 0.09 s faster.

0.2

0 0.2 0.4 0.6 0.8 1 1.2

Real time
(s)
© SOFAD / All Rights Reserved.

You may now fill in the self-evaluation grid for the five targeted criteria. Refer to the grid at the end of the guide. Your teacher
or tutor may also provide you with the evaluation indicators for this LES. These will help you judge the quality of your solution.

159
 TABLE OF
CONTENTS

CHAPTER 2
SITUATION 2.1 THE PRODIGY ACQUISITION 2.1 A PAGES 69 TO 77
EXPLORATION 2.1 PAGES 67 AND 68 12x
____
1 a) ​
​ ​5 4x​
3
24a 24 3 ​a​1 2 2​ ​ a​​21​ 2___
2 1
1 The first audience member suggests an n of 7. ____
​ 2 ​5 ________
b) ​ ​ ​5 ___
​ ​5 ​ ​
The expression therefore must be solved for n 5 7. 8​a​ ​ 8 2 2a
14​y​3​ 14​y​3 2 3​
____
n 1 1 _____
_____ 1 2 (7) 1 1 ______ 1 2 ​ 3 ​5 ______
c) ​ ​ ​5 2​y​0​5 2​
​ ​  ​ 1 ​   ​ 1 ________
​   ​ 5 ______
​   ​ 1 ​   ​ 1 ​ __________ ​ 7​y​ ​ 7
n n 1 2 n(n 1 2) (7) (7) 1 2 (7)((7) 1 2)
8 1 ___ 2 2 If x is the unknown value, the statements may be translated
5 ​__​1 __ ​ ​1 ​ ​
7 9 63 as follows:
72 1 7 1 2
5 ​_______​ a) ​
x
__
​​
x11
______
b) ​
​ ​ c) ​
1
x 1 ___
​ ​ d) ​
​
x
____​
63 2 2x 2x x12
81
5 ​___​
63 3 a) 
No restrictions, because there is no variable in the
9
5 ​__​ denominator.
7
b) The restriction is x  0, because x2  0.
2 a) When n 5 1, the final value is 3. 22x 21
____
​ 2 ​​ 5 ___
​​  ​​
12x 6x
b) When n 5 2, the final value is 2.
c) The restrictions are y  0 and x  22, because 6y  0
Solution to The Prodigy and x 1 2  0.
Number Resulting Simplified 3​y​​
________ ​1​​
chosen fraction answer ​​   ​​ 5 ______
​​   ​​.
6y(x 1 2) 2x 1 4
3
__
1 ​ ​ 3 1 1 133 1 3 1 4 2
__
1 ​ ​1 __
4 a) ​ ​ ​​5 ​______​1 __
​ ​ 5 __
​ ​1 __
​ ​ 5 __
​ ​ 5 __
​​
This book is the property of Dickson Joseph.

4 2 6 233 6 6 6 6 3
2 __
​ ​ 2 2 3 ______
__ 2 3 4 ______
3 3 5 ___ 8 15 23
2 ​ ​1 __
b) ​ ​ ​​5 ​ ​1 ​ ​ 5 ​ ​1 ___
​ ​5 ___
​ ​
5
__ 5
__
5 4 5 3 4 4 3 5 20 20 20
3 ​ ​ ​ ​
c) When working with algebraic expressions like __
​​  2x ​ 1 __
1 5
3 3 ​  6 ​​,
6
__ 3
__ the same procedure is used as in arithmetic.
4 ​ ​ ​ ​
4 2
7
__ 7
__ Explanations Algebraic procedure
5 ​ ​ ​ ​
5 5 Determine the common 1 5 133 53x
___
​ ​1 __
​ ​​5 ______
​ ​1 _____
​ ​
8
__ 4
__ denominator; in this case, 6x.
6 ​ ​ ​ ​ 2x 6 2x 3 3 6 3 x
6 3 If it is an algebraic fraction, name
the restriction(s). Restriction: x  0
9
__ 9
__
7 ​ ​ ​ ​
7 7 3
___ 5x 3 1 5x
Add the transformed fractions. ​ ​1 ___
​ ​5 _____
​ ​
6x 6x 6x
3 By comparing the initial number and the denominator of Simplify the result, if possible.
3 1 5x
the resulting fraction, you can see that they are equal. In the case of an algebraic ​_____​​, where x  0
fraction, note the restriction(s). 6x

7
___ 3 7 333 7 9 2 1
4 You can conclude that the numerator is always equal to ​ ​2 __
5 a) ​ ​ ​​5 ___
​​ ​2 _____
​ ​ 5 ___
​ ​2 ___
​ ​​5 2___
​ ​ 5 2___
​​
2 more than the initial number. 12 4 12 4 3 3 12 12 12 6
3
___ 2a 1
​ ​2 ___
​ ​2 ___

© SOFAD / All Rights Reserved.

b) ​ ​ ​
n11 1 2 5a 3 2a
5 Considering the expression ____
​ ​1 ____
​ ​1 ______
​ ​5 ?​,
n n 1 2 n(n 1 2) Explanations Algebraic procedure
it seems that, in light of the previous examples, regardless of Determine 3
___ 2a
​ ​2 ___
​ ​2 ___
1
​ ​​5
the common 5a 3 2a
the numerical value attributed to the variable n, the result is
denominator; 336 2a 3 10a 1 3 15
always equal to ____
n12
​​  n ​​. That is, 2 more than the number in the in this case, 30a. ​______​2 ​________​2 ​_______​
5a 3 6 3 3 10a 2a 3 15
numerator position and exactly the value of the number in If it is an algebraic
fraction, name the Restriction: a  0
the denominator position.
restriction(s).
18 20​a​2​ ____
15 18 2 20​a​2​2 15
Subtract the ​____​2 _____
​ ​2 ​ ​​ 5 ​​___________​
30a 30a 30a 30a
transformed 3 2 20​a​2​
fractions. 5 ​​_______​
30a
Simplify the result,
if possible.
In the case of an 3 2 20​a​2​
_______
​ ​​, where a  0
algebraic fraction, 30a
note the
restriction(s).

160 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

ANSWER KEY
6 a) In this algebraic expression, the common denominator
VISUAL EXPLANATION:
CHAPTER 2
is x(x 1 4), and the restrictions are x  0 and x  24.
This square may also be viewed as an assembly of
2
____ x 1 2 ______8 four quadrilaterals.
​ ​1 ____
​ ​1 ​ ​
x14 x x(x 1 4)
a b
23x (x 1 2)(x 1 4) ______8
5 ​________
​ ​1 _________
​ ​1 ​ ​
(x 1 4) 3 x x(x 1 4) x(x 1 4) a a2 ab
2x ​x​2​1 4x 1 2x 1 8 8
5 ​______
​ ​1 ​___________​1 ______
​ ​
x(x 1 4) x(x 1 4) x(x 1 4)
b ab b2
x​ ​2​1 8x 1 16
5 ​_________
​ ​, where x  0 and x  24
x(x 1 4)
The area of each quadrilateral is indicated in the figure below.
b) In this algebraic expression, the common denominator You can see that these areas are equivalent to the square of each
25
is y(2y 1 5), and the restrictions are y  0 and y  __
​​   ​​. of the terms of (a 1 b) and twice the product of the two terms.
2
Thus, the area of the square is equivalent to the sum of these
This second restriction comes from solving the
four parts: a2 1 ab 1 ab 1 b2 is equivalent to a2 1 2ab 1 b2.
equation 2y 1 5  0.
Therefore, the equation (a 1 b)2 5 a2 1 2ab 1 b2 is an identity.
5 30 2 4y
__
​ ​2 ______
​ ​ This expression (a 1 b)2 is called the square of a binomial.
y 2y 1 5
5(2y 1 5) 2 y(30 2 4y)
​5 ​________________
9 Since the identity indicates that the square of a binomial is

ANSWER KEY
CHAPTER 2
   ​
y(2y 1 5) equal to the sum of the square of each term and the double
10y 1 25 2 30y 1 4​y​ ​ 2 of the product of the two, you get:
​5 ​_______________
   ​
y(2y 1 5)
a) (x 1 8)2 5 x2 1 16x 1 64
4​y​ ​2 20y 1 25
2 25
​5 ​___________​, where y  0 and __
​​   ​​ b) (x 2 2y)2 5 x2 2 4xy 1 4y2
y(2y 1 5) 2

c) (2xy 2 3)2 5 4x2y2 2 12xy 1 9

7 a) Validation with x 5 2:
d) (x2 1 2x)2 5 x4 1 4x3 1 4x2
2
____ x 1 2 ______8 ​x​2​1 8x 1 16
​
​ ​1 ____
​ ​1 ​ ​​5 _________
​ ​
x14 x x(x 1 4) x(x 1 4)
10 a) The square root of 4x2 is 2x.
2
_____ (2) 1 2 ________8 ​(2)​​2​1 8(2) 1 16
​ ​1 _____
​ ​1 ​ ​​ 5 ___________
​ ​
(2) 1 4 (2) (2)((2) 1 4) (2)((2) 1 4) b) The square root of 25 is 5.

2 4 ___
__ 8 4 1 16 1 16 c) By doubling the product of the two terms; that is,
​ ​1 __
​ ​ ​1 ​ ​​5 ________
​ ​
6 2 12 12 2 3 2x 3 5 5 20x.
353

b) Validation with y 5 2:
11 The perfect squares are a, c and d:
a) x2 2 6x 1 9 5 (x 2 3)2
5 30 2 4y ___________
__ 4​y​2​2 20y 1 25
​ ​2 ______
​ ​ ​​5 ​ ​
y 2y 1 5 y(2y 1 5) b) It is not a perfect square, because the middle term is not
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5 30 2 4(2) _____________
4​(2)​​ ​2 20(2) 1 25
2 double the product of the roots of the first and third
​___​2 _______
​ ​​ 5 ​   ​
(2) 2(2) 1 5 (2)(2(2) 1 5) terms. As the roots are x and 2, for it to be a perfect
square, the middle term should have been 2(2x) 5 4x.
5
___ 30 2 8 _________
16 2 40 1 25
​ ​2 _____
​ ​ ​​5 ​ ​
(2) 415 (2)(4 1 5) c) 4x2 1 12x 1 9 5 (2x 1 3)2
1
___ 1
​ ​​5 ___
​ ​ ​ d) 36x2 2 84x 1 49 5 (6x 2 7)2
18 18

8 The area of the square is (a 1 b)2.

Using multiplication, you get:

(a 1 b)(a 1 b) 5 a2 1 ab 1 ab 1 b2 5 a2 1 2ab 1 b2

161
 TABLE OF
CONTENTS

ANSWER KEY EXPLANATION:

​
2x
______
14 a) ​ ​1 _____
​
x 1 1.5
​
6
2x 1 3
CHAPTER 2 root of 49 may be 7 or
The square 27. In this case,
​
2x
​5 ______
6
​1 ________
​ ​
a negative square root is the option to choose. x 1 1.5 2(x 1 1.5)
2(2x) 6
• The square roots of the first and third terms ​5 ________
​ ​1 ________
​ ​
2(x 1 1.5) 2(x 1 1.5)
are 6x and (27).
4x 6
• The middle term is 2 3 6x 3 27 5 284x. ​5 _____
​ ​1 _____
​ ​
2x 1 3 2x 1 3
4x 1 6 _______
2(2x 1 3)
12 a) x2 1 10x 1 25 ​5 _____
​ ​​5
​ ​ ​
2x 1 3 2x 1 3
b) 4x2 1 12xy 1 9y2 or 4x2 2 12xy 1 9y2 5 2, where x  21.5

6x 3​x​2​2 1
c) 25x2 2 30x 1 9 b) ​​_____​2 ___________
​ ​
3x 2 1 (2x 2 1)(3x 2 1)
13 a) • 6x(2x 2 1) 2 (3​x​2​2 1)
The root of the first term is 8x. ​5 ​_______________
     ​
(2x 2 1)(3x 2 1)
• The root of the third term is 1.
12​x​2​2 6x 2 3​x​2​1 1
• Double the product of the two roots is 16x. ​5 ​______________
    ​
(2x 2 1)(3x 2 1)
• This product matches the middle term. Therefore, this 9​x​2​2 6x 1 1
​5 ___________
​ ​
perfect square trinomial is equivalent to (8x 1 1)2. (2x 2 1)(3x 2 1)
​(3x 2 1)​​2​
b) By replacing the perfect square trinomial with its ​5 ___________
​ ​
(2x 2 1)(3x 2 1)
factored equivalent, you get _______
(​ 8x 1 1)​2​
​​  8x 1 1 ​​. This can then
3x 2 1 1 1
be simplified to get 8x 1 1, where x  2​__
1
​ ​.​ ​5 _____
​ ​​, where x  __
​ ​​and x  __
​​
8 2x 2 1 3 2
This book is the property of Dickson Joseph.

SOLUTION 2.1
The Prodigy PAGES 78 AND 79
Sample solution:

Conjecture

It seems that, in light of the examples provided, regardless of the numerical value attributed to the variable n, the result is always equal
n12
to _____
​​   ​ ​. That is, 2 more than the number in the numerator position and exactly the value of the number in the denominator position.
n
Simplifying the Expression
n11 1 2
First, the following expression must be simplified: _____
​​   ​ 1 _____
​   ​ 1 _______
​   ​​.
n n12 n(n 1 2)
The common denominators are n and (n 1 2); the restrictions are n  22 and n  0.

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n11 1 2
​____​1 ____
​ ​1 ______
​ ​ This algebraic expression is the same as the conjecture that was
n n 1 2 n(n 1 2)
formulated.
(n 1 1)(n 1 2) 13n 2
​5 ​__________​1 ______
​ ​1 ______
​ ​ n 1 1 ____
Thus, you can state that _____
​​   ​ 1​
1
​
2
​1 ______
n12
​5 ____
​ ​​.
n(n 1 2) n(n 1 2) n(n 1 2) n n 1 2 n(n 1 2) n
(n 1 1)(n 1 2) 1 n 1 2 Procedure
​5 ​______________
   ​
n(n 1 2)
Given that the expression may be simplified to ____
n12
​​  n ​​, the prodigy
​ ​2​1 3n 1 2 1 n 1 2
n only has to add 2 to the initial number and divide the result by
​ _____________
5 ​   ​
n(n 1 2) the initial number. He doesn’t have to perform a long series of
​n​2​1 4n 1 4 calculations based on the initial algebraic expression. Instead,
​5 ________
​ ​
he simply solves this short expression: ____
n12
n(n 1 2) ​​  n ​​ . However, the
chosen number cannot be 2 or 0—otherwise, it would be
2
​(n 1 2)​​2​
​5 ______
​ ​ impossible to solve.
n(n 1 2)
n12
​5 ____
​ ​
n

162 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

ANSWER KEY
ACQUISITION 2.1 B PAGES 80 TO 82
CHAPTER
1 a) (a 1 b)(a 22b) 5 a 1 ab 2 ab 2 b 2 2
4 a) (4x 2 1)(4x 1 1)
5a 2b 2 2
b) (6 2 2a)(6 1 2a)
b) Using the distributive property, you obtain a polynomial
c) 225 1 x2 5 x2 2 25
with four terms, the two middle terms of which cancel
5 (x 2 5)(x 1 5)
each other out. After simplification, all that remains is a
binomial, which is in fact the difference of two squares.
5 The length of each of the sides of the rectangle can be
represented by (8x 2 5) and (8x 1 5), because the area is
2 a) (x 1 3)(x 2 3) 5 x2 2 3x 1 3x 2 9 5 x2 2 9
equal to the product of the length of the rectangle’s sides
b) (4x 1 y)(4x 2 y) 5 16x2 2 4xy 1 4xy 2 y2 5 16x2 2 y2 and is represented by the difference of two squares.

3x 1 1
______ 3x 1 1
3 a) The result of simple factorization of the denominator is 6 a) ​
​ 2 ​ 5 ___________
​ ​
9 ​x​ ​2 1 (3x 2 1)(3x 1 1)
3x(x 2 3). Therefore, the restrictions are: x  0 and x  3.
1 1 1
5 ______
​​   ​​, where x  __
​ ​​and x  2​__
​ ​​
b) The numerator can be factored since it is the difference 3x 2 1 3 3
of two squares. It becomes (x 1 3)(x 2 3).
1.44 2 ​a​2​ _____________
_______ (1.2 2 a )(1.2 1 a)
b) ​
​ ​ 5 ​   ​
You therefore obtain the following identity: a 1 1.2 a 1 1.2
​x​2​2 9
_________ (x 1 3)(x 2 3) 51​ .2 2 a​, where x  21.2
​ 2 ​​5 _____________
​   ​

ANSWER KEY
CHAPTER 2
3​x​ ​2 9x 3x(x 2 3)
By factoring the expressions, the following simplification
is possible:
(x 1 3)​(x 2 3)​ ______
____________ x13
​​     ​​ 5 ​ ​​, where x  0 and x  3
3x​(x 2 3)​ 3x

CONSOLIDATION 2.1 PAGES 83 TO 85 3 The restrictions are identified by dividing by zero. They are
therefore determined by finding the values that produce a
1 a) 522 5 (50 1 2)2 5 502 1 2 3 (50 3 2) 1 22 value of 0 in the denominator.
5 2500 1 200 1 4 5 2704
a) x 1 1  0, therefore x  21
b) 39 5 (40 2 1)2 5 402 1 2 3 (40 3 21) 1 12
2
1
b) 2x 2 1  0, therefore x  __
​​
5 1600 2 80 1 1 5 1521 2
c) 3x  0 and x 2 2  0, therefore x  0 and x  2
c) 2432 2 2422 5 (243 1 242) 3 (243 2 242) 5 485 3 1
5 485 d) 2x 2 6  0, therefore x  3
d) 61 3 59 5 (60 1 1)(60 2 1) 5 60 2 1 5 3600 2 1 2 2
1 3 2a a
5 3599
4 a) 5​______
​ ​1 ___
​ 2​
a 3 2a 2​a​ ​
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2a 1 a
e) 63 3 57 5 (60 1 3)(60 2 3) 5 602 2 32 5 3600 2 9 5​_____
​ ​
2​a​2​
5 3591
3a
​5 ___
​ 2​
2​a​ ​
2 a) If x 5 21, the equation becomes:
3
3
______ 2(21) 2 1 ___________
1 2(21) ​5 ___
​ ​​, where a  0
​2 ​1 ​_______ ​2 ​ 2 2 ​5 ______
​ ​ 2a
( 1) 2 1 (21) ( 1)(( 1) 2 1) (21) 2 1
3 1 2 3(x 1 1)
5 ​_________​
2
​ ___
​2 ​1 3 2 __
​ ​5 ___
​ ​ b) ​
2 2 22 (x 2 1)(x 1 1)
151 3
​5 ____
​ ​​, where x  21 and where x  1
1 x21
b) If x 5 __ ​ ​​, the equation becomes:
2
_______
​ 
3
 ​ 1 ​​ 
()
2​ __
1
​  2 ​ ​2 1
________  ​​ 2 ​  ___________ 1
 ​ 5 ​ 
2​ __
_______ ()
1
​  2 ​ ​
 ​
c) ​ ​
​(x 1 2)​​2​
5 __________
(x 2 2)(x 1 2)
​

(2)
​ __
1
​  ​ ​2 1 (2)
​ __
1
​  ​ ​ (2)((2) ) (2)
1 1
​  ​ ​​ ​ ​ __​ ​2 1 ​ ​ __
​ __
1
​  ​ ​2 1 x12
​5 ____
​ ​​, where x  22 and where x  2
3 0 1 1 x22
​​ ___
1
 ​ 1 __
​  1  ​ 2 ___
​  1  ​ 5 ___
​  1  ​​
2__
​ ​ __ ​  ​ 2__ ​ ​ 2__ ​​
2 2 4 2
26 1 0 1 4 5 22
22 5 22

163
 TABLE OF
CONTENTS

ANSWER KEY
d) In this algebraic expression, the common denominator is c) Given that the middle term is double the product of the
x(3x 1 4) and the restrictions are x  0 and x  ___
24

CHAPTER 2
12x(x) 2 (x 2 5)(3x 1 4) 1 13x 2 4
_______________________
​​  3 ​​. two terms of the binomial and the last term is the square
of the second term of the binomial, the ___following
5 ​   
  ​
x(3x 1 4) equation can be established: 2 3 a 3 ​√ 25 ​5 250x.
12​x​2​2 (3​x​2​1 4x 2 15x 2 20) 1 13x 2 4 By isolating, we find that the value of a is 5x; squared
5 ​___________________________
      ​
x(3x 1 4) it is 25x2: 25x2 2 50x 1 25.
12​x​2​2 3​x​2​2 4x 1 15x 1 20 1 13x 2 4
5 ​__________________________
    ​
x(3x 1 4) 8 Since each of these is a difference of two squares, factoring is
9​x​2​1 24x 1 16 ​(3x 1 4)​​  ​2​​ ______3x 1 4 achieved by determining the square root of each term. The
5 ​___________​ 5 ________
​​   ​​ 5 ​​  ​​
x(3x 1 4) x(​3x 1 4​) x product will be the sum and the difference of these terms.

5 • For each square of a binomial, the resulting trinomial a) (3x 1 4)(3x 2 4)

is always made up of the square of the first term, the
b) (x 1 2y)(x 2 2y)
double of the product of the two terms and the square
of the second term. c) (xy 1 5)(xy 2 5)
• For each product of a sum and the difference of these d) (x2 1 1)(x2 2 1) 5 (x2 1 1)(x 1 1)(x 2 1)
same terms, the resulting binomial is the difference
between the squares of the two terms. In d), after factoring once, you can see that one of the factors
is also the difference of two squares. This factor can then be
a) x2 1 16x 1 64 factored to get a product of three binomials.
b) 25x2 2 10x 1 1 x(x 1 2) ​x​(x 1 2) x12
9 a) 5 ​________​5 _______
​​   ​​ 5 ______
​ ​, where x  0
c) x 2 25
2 2​x​ ​
2 2
2x​ ​ 2x
This book is the property of Dickson Joseph.

3(2x 1 3) 3​(2x 1 3)​ ___ 3 3

d) x2 2 4xy 1 4y2 5 ​__________​5 _________
b) ​ ​  ​ __
 ​ 5 ​ ​, where x  0 and x  2 ​​
2x(2x 1 3) 2x​(2x 1 3)​ 2x 2
e) 4x2y2 2 12xy 1 9 (x 1 2)(x 2 2) ​(x 1 2)​(x 2 2) ______
x22
5 _____________
c) ​ ​   ​5 ____________
​     ​ 5 ​ ​​,
f ) 4x2 2 y2 x(x 1 2) x​(x 1 2)​ x
where x  0 and x  22
g) x4 1 4x3 1 4x2
​(x 2 1)​​2​ ​(x 2 1)​​​ ​2​​ x21
d) 5 _____________
​  ​5 ____________
​​     ​​ 5 ______ ​ ​,
h) 9 2 25x4 (x 1 1)(x 2 1) (x 1 1)​(x 2 1)​ x 1 1
where x  21 and x  1
6 To check that a trinomial is a perfect square, you must 2x(2x 1 1) _________ 2x​(2x 1 1)​ _______ 2x 1
determine whether the middle term is double the product of 5 __________
e) ​ ​ ​5 ​   ​ 5 ​ ​ __
​​, where x  2 ​​
​(2x 1 1)​​2​ ​(2x 1 1)​​  ​2​​ 2x 1 1 2
the square roots of the two other terms. The second one is 1 x11
not a perfect square because the middle term should be 8x. 10 a) 5 _____________
​  ​​1 ​______​
(x 1 1)(x 2 1) x21
The fifth one is not a perfect square because the middle 1 ​(x 1 1)​​2​
term should be 212x. 5 _____________
​  ​​1 ​_____________
  ​
(x 1 1)(x 2 1) (x 1 1)(x 2 1)
1 1 ​x​2​1 2x 1 1
x2 2 14x 1 49 (x 2 7)2 5 _________________
​  
  ​
(x 1 1)(x 2 1)

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x2 1 4x 1 16 ​x​2​1 2x 1 2
5 ____________
​ 2 ​
4x2 120x 1 25 (2x 1 5)2 ​x​ ​2 1
b) Simply take the first steps from a) but change the
36x2 2 6x 1 0.25 (6x 2 0.5)2 operation sign.
1 x11
4x2 2 24x 1 9 ​5 ​_______ ​​2 ______
​ ​
​x​2​ 2 1 x 2 1
9x2 1 12x 1 4 (3x 1 2)2 5 (…)
1 2 (​x​2​1 2x 1 1)
7 a) Since the middle term is double the product of the two 5 _________________
​    ​
(x 1 1)(x 2 1)
terms, it can be found using the following equation: 2 ​x​2​2 2x

2 3 x 3 b 5 18x. Thus, the value of b is 9; squared it is 5 _________

​ 2 ​
​x​ ​2 1
​
81: x2 1 18x 1 81. x11 ​​(x 1 ​
1)​​​ 1
5 __________
c) ​ ​ 2 ​5 ​ ___________________
   ​ 5 ________
   ​ ​
b) Using the square root of the first and last terms (​x​ ​2 1)(x 2 1) ​(x 1 1)​(x 2 1)(x 2 1) ​(x 2 1)​​2​
​
(3x and 10y), you can determine the middle term by
doubling the product of the two: 9x2 1 60xy 1 100y2
d) ​ (
​2
1
5 ​ ________ )( )
x21
​ ​ ______
​
​x​ ​2 1 x 1 1
​​(x 2 ​
1)​​​
​ ​5 ___________________
   ​ 5 ________
​    ​
1
​
(x 1 1)​(x 2 1)​(x 1 1) ​(x 1 1)​​2​
or 9x2 2 60xy 1 100y2.

164 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

ANSWER KEY
SITUATION 2.2
CHAPTER 2
THE TELEPATH
EXPLORATION 2.2 PAGES 87 AND 88
1 a) Triple of 3 a) Secret number 2
Cube the square
1. Calculate the cube of the secret number. 8
64 48 2. Add the triple of its square. 8 1 12 5 20
3. To the result, add three more than the
20 1 5 5 25
secret number.
Number
Three more Two less 4. Multiply the answer by two less than the
than the 7 4 2 than the 25 3 0 5 0
secret number.
number number
5. Divide the answer by one more than the 0
__
​ ​​5 0
square of the secret number. 5
17 20 6. Subtract the answer from the sum of the
620
secret number and its square.
One more than Sum of the 7. Calculate the cube of the secret number. 6
the square of number and
the number the square
b) Following the telepath’s instructions, it seems that the
answer is always 6 regardless of the number chosen.
(64 1 48 1 7 ) 3 2
b) The expression is 20 2 _____________
​   ​​.
17 4 Your expression should look like this:

ANSWER KEY
CHAPTER 2
EXPLANATION: (​x​3​1 3​x​2​1 (x 1 3))(x 2 2)
​(​x​2​1 x) 2 ___________________
​ ​
​x​2​1 1
Secret number 4
1. Calculate the cube of the EXPLANATION:
64
secret number.
2. Add the triple of its square. 64 1 48 Secret number x
3. To the result, add three more 1. Calculate the cube
64 1 48 1 7 of the secret x3
than the secret number.
number.
4. Multiply the answer by two
(64 1 48 1 7) 3 2 2. Add the triple of its
less than the secret number. x3 1 3x2
square.
5. Divide the answer by one
(64 1 48 1 7) 3 2 3. To the result, add
more than the square of the ​_____________
   ​
secret number. 17 three more than the x3 1 3x2 1 (x13)
secret number.
6. Subtract the answer from the
(64 1 48 1 7) 3 2 4. Multiply the answer
sum of the secret number 20 2 ​_____________
​   ​​
and its square. 17 by two less than the (x3 1 3x21 (x 1 3))(x 2 2)
secret number.
5. Divide the answer
by one more than (​ ​x​3​1 3​x​2​1 (​ x 1 3))​ (​ x 2 2)​
​___________________
  ​
the square of the ​x​2​1 1
(64 1 48 1 7 )  3 2 secret number.
2 20 2 ​​_____________
   ​​
17 6. Subtract the answer
​​119 3 2 ​​
5 20 2 _______
from the sum of the (​ ​x​3​1 3​x​2​1
(x2 1 x) 2 ___________________
​​
(​ x 1 3))​ (​ x 2 2)​
  ​​
17 secret number and ​x​2​1 1
© SOFAD / All Rights Reserved.

its square.
238 ​
____
5 20 2 ​ ​ 5 20 2 14 5 6
​
17
As was the case for the five volunteers, the resulting
number is 6.

ACQUISITION 2.2 A PAGES 89 TO 93

1 To simplify these expressions, you must apply what you 2 a) and b) To multiply the binomial by the trinomial, you
know about the laws of exponents and the distributive must carry out six multiplications of monomials. The final
property. result will therefore contain a maximum of six terms, and the
degree will be equal to the sum of the factors’ degrees;
a) 6x2y5 3 (22x3y) 5 212x5y6
that is, 2 1 2 5 4.
b) 7x(2x 2 3) 5 14x2 2 21x
5 (x2 2 4x)(x2 1 2x 17)
c) (5x2 1 3)(x 2 4) 5 5x3 2 20x2 1 3x 2 12
5 x4 1 2x3 1 7x2 2 4x3 2 8x2 2 28x
5 x4 2 2x3 2 x2 2 28x

165
 TABLE OF
CONTENTS

ANSWER KEY
3 a) (9x5 1 3)(2x3 2 5x 1 2) 5 18x8 2 45x6 1 18x5 1 6x3 2 15x 1 6
CHAPTER 2
b) (3a2 2 2a 2 5)(a3 2 a 1 2) 5 3a5 2 3a3 1 6a2 2 2a4 1 2a2 2 4a 2 5a3 1 5a 2 10
5 3a5 2 2a4 2 8a3 1 8a2 1 a 2 10

EXPLANATION:
It is always preferable to write a simplified polynomial according to the order of the terms’ degrees: 3a5 2 2a4 2 8a3 1 8a2 1 a 2 10.

c) (0.5y3 1 0.25)(y4 2 3y21 6y 1 4) 5 0.5y72 1.5y5 1 3y4 1 2y3 1 0.25y4 2 0.75y2 1 1.5y 1 1
5 0.5y72 1.5y5 1 3.25y41 2y3 2 0.75y2 1 1.5y 1 1

4 When dividing powers, it is important to remember to use 6 a) 1250 5 b) 6936 12

the laws of exponents. In this case, the coefficients must be 2 10 250 2 60 578
divided and the exponents must be subtracted from the 25 93
powers with the same base. 2 25 2 84
0 96
4​x​3​
____
a) ​
​ ​​5 2x2 20 2 96
2x
0 0
3e
____ 1
​ 4 ​​5 ___
b) ​ ​ ​
6​e​ ​ 2​e​3​ 7 a) Personal answer.
x​y​4​ ​y​3​
____
​ 2 ​​5 __
c) ​ ​ ​ b) Sample answer:
​x​ ​y x
This book is the property of Dickson Joseph.

4 ​x​ ​ 2x2
1 To calculate 2x2 2 7x 2 4 x 2 4, you must proceed as
5 a) (4x2 2 2x) 4 4x​ 5 ___
​ ​2 ___ ​ x 2 __
​ ​​5 ​​ you would to divide numbers like 1250 5. However,
4x 4x 2
8 ​a​2​ ___
___ 4a __
1 you must go through each member of the binomial one
b) (8a 2 4a 11) 4 2 ​5 ​ ​2 ​ ​1 ​ ​
2
2 2 2 by one rather than treating it as a whole. This kind of
1
​ 5 4 ​a​2​2 2a 1 __
​​ algebraic division is performed by repeating a certain
2 number of steps. These steps will be described later on
c) (27x4 1 6x3 2 3x2 2 3x) 4 22x in the acquisition activity.
27​x​4​ 6​x​3​ 3x 2 3x
​5 ​_____
22x
​1 ____
​2 ​2 ____​2 ​ ​2 ____
​2 ​
2x 2x 2x
7​x​3​
___ 3x __
___ 3
​5 ​ ​2 3 ​x​ ​1 ​ ​1 ​ ​
2
2 2 2

8 a) To guide you through this first division, a highly detailed solution is provided.

Find a term that, when multiplied by x, gives 2x3. In this case, 2x2 is the first term of the quotient.
By multiplying 2x2 by x 1 2, you get 2x3 1 4x2, which is written below the dividend,
in the appropriate columns. 2x3 1 5x2 1 7x 1 10 x 1 2
2x3 1 4x2 2x2

© SOFAD / All Rights Reserved.

This means that x 1 2 goes into 2x3 1 4x2 a total of 2x2 times. Now you must see what remains
of the dividend.

Since this binomial must be subtracted from the dividend, it is preceded by the minus sign and
2x3 1 5x2 1 7x 1 10 x 1 2
surrounded by brackets to clearly show that both terms in the binomial must be subtracted.
2 (2x3 1 4x2) 2x2
NOTE: The result of 2x3 2 2x3 is 0; the x3 term disappears. All that remains is the result of
5x2 2 4x2, which is x2. x2

Once the subtraction is finished, bring down the next term (7x) and start the process all over again. 2x3 1 5x2 1 7x 1 10 x 1 2
You now need to find a term that, when multiplied by x, gives x2. In this case, x will be the second term 2 (2x3 1 4x2) 2x2 1 x
of the quotient. x2 1 7x
The product of x and x 1 2 gives x2 1 2x, which is then subtracted. 2 (x2 1 2x)
The result of this second subtraction is 5x, since the x2 terms cancel each other out. 5x

2x3 1 5x2 1 7x 1 10 x 1 2
2 (2x3 1 4x2) 2x2 1 x 1 5
The process continues by bringing down the term 10. x2 1 7x
By adding the constant 5 as the third term of the quotient, the multiplication of 5 and x 1 2 means 2 (x2 1 2x)
that subtracting the two binomials gives 0. This is always the result of division without a remainder. 5x 1 10
2 (5x 1 10)
0
(2x3 1 5x2 1 7x 1 10) 4 (x 1 2)
Write the solution.
5 2x2 1 x 1 5

166 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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ANSWER KEY
b) 2x2 2 7x 2 4 x 2 4 9 A rational expression may be simplified using the division of
CHAPTER 2
2 (2x 2 8x)* 2
2x 1 1
polynomials. Remember to put the polynomial in the correct
order before performing the division.
x24
2 (x 2 4) 4a4 1 4a3 2 a 2 1 4a3 2 1
0 2 (4a4 2 a)* a11
Therefore, (2x 2 7x 2 4) 4 (x 2 4) 5 2x 1 1.
2 4a3 21

*C
 aution! You must be careful to carry out the 2 (4a3 2 1)*
subtraction of negative numbers correctly. 0
For example, 27x 2 (28x) 5 27x 1 8x 5 x. The simplified expression gives a 1 1.

SOLUTION 2.2 ACQUISITION 2.2 B PAGES 96 TO 99

The Telepath PAGES 94 AND 95 1 a) 127 7
2 7 18
You need to prove the following conjecture: 57
No matter the value of x, 2 56
1
(​x​3​1 3​x​2​1 (x 1 3))(x 2 2)
​(​x​2​1 x) 2 ______________________
​ ​​5 6
​x​2​1 1
EXPLANATION:

ANSWER KEY
CHAPTER 2
1) Multiply the four-term polynomial by the binomial (x 2 2):
When carrying out the division algorithm, you will see that
(x3 1 3x2 1 x 1 3)(x 2 2) you have the value 1 left over at the end.
5 x4 2 2x3 1 3x3 2 6x2 1 x2 2 2x 1 3x 2 6 1 1
The quotient is therefore 18​​__​​. The ​__​represents the
5 x4 1 x3 2 5x2 1 x 2 6 7 7
remainder of 1 over the divisor of 7.
1
2) Divide the polynomial by the binomial (x2 1 1): Thus, 127 4 7 5 18​__
​ ​.
7
x4 1 x3 2 5x2 1 x 2 6 x2 1 1
2 (x4 x2) x2 1 x 2 6
NOTE: Writing “127 4 7 5 18 with remainder of 1” is a
x 2 6x 1 x
3 2
mistake, since “remainder of 1” is not a mathematical
2 (x3 1 x)
26x2
operation. An equality must contain nothing but numbers
26
and operation signs, and perhaps brackets.
2 (26x2 2 6)
0
The result is the expression x2 1 x 2 6. b) 25,348 15
2 15 1689
NOTE: Note that, in the first multiplication of the quotient 103
term by the binomial, no x3 term appears. But for the following 2 90
134
subtraction, it is essential to bring this third-degree term down
2 120
to ensure the division algorithm continues.
148
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2 135
3) 
Subtract the result from the sum of the number and its square:
13
x2 1 x 2 (x2 1 x 2 6) 5 x2 1 x 2 x2 2 x 1 6 5 6 13
Thus, 25,348 4 15 5 1689​___
​ ​​.
Since there are no more variables in the final expression, 15
the result will always equal 6.
2 a) 2x3 1 5x2 1 7x 1 6 x 1 2
Another possible procedure:
2 (2x3 1 4x2) 2x2 1 x 1 5
In the conjecture to be proven, the term (x2 1 x) has a x 1 7x
2

denominator equal to 1. You can then carry out the subtraction 2 (x2 1 2x)
by using x2 1 1 as the common denominator, then simplify the 5x 1 6
expression to get the result of 6. 2 (5x 1 10)
24
Conclusion
The telepath has not proven that she has a gift. In fact, in this The division gives 2x2 1 x 1 5, remainder 24.
exercise, by applying mathematical reasoning to the algebraic
expression, you have demonstrated that the result is always 6
regardless of the initial number.

167
 TABLE OF
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ANSWER KEY 24
b) 2x2 1 x 1 5 1 ____
​
x12
​ b) 3x3 2 6x2 1 0x 1 10 x 2 1
CHAPTER 2
EXPLANATION:
2 (3x3 2 3x2)
2 3x2 1 0x
3x2 2 3x 2 3

2 (23x2 1 3x)
Since the remainder is 24 and the divisor is X 1 2, 23x 1 10
you can represent the remainder as a fraction, 2 (23x 1 3)
as is done in arithmetic division: 7
24
(2x3 1 5x2 1 7x 16) 4 (x 1 2) 5 (2x2 1 x 1 5) 1 ______
​ ​ The result can be written as follows:
x12
7
(3x3 2 6x2 1 10) 4 (x 2 1) 5 3x2 2 3x 2 3 1 ____
​ ​
x21
Remainder
_______
Dividend Divisor Quotient ​
​ ​
Divisor 4 Just as an arithmetic division can be checked by performing
the reverse multiplication (8 4 2 5 4, so 2 3 4 5 8), an
3 a) 8a3 1 22a2 2 21a 2 3 2a 1 7 algebraic division can also be checked by performing the
2 (8a3 1 28a2) 4a2 2 3a reverse multiplication. Therefore:
26a2 2 21a

2 (26a2 2 21a) (2x2 2 7x 2 4) 4 (x 2 4) 5 2x 1 1 implies that

02 3 (x 2 4)(2x 1 1) must equal 2x2 2 7x 2 4.

The result can be written as follows: (x 2 4)(2x 1 1) 5 2x2 1 x 2 8x 2 4 5 2x2 2 7x 2 4

23
(8a3 1 22a2 2 21a 2 3) 4 (2a 1 7) 5 4a2 2 3a 1 _____
​ ​ Thus the quotient is correct.
2a 1 7

5 As in question 4, this division can be checked using the reverse multiplication. Therefore:
This book is the property of Dickson Joseph.

( )
24 24
(2x3 1 5x2 1 7x 1 6) 4 (x 1 2) 5 2x2 1 x 1 5 1​____
​ ​ implies that (x 1 2)​​ 2​x​2​1 x 1 5 1 ____
​ ​ ​​must equal 2x3 1 5x2 1 7x 1 6.
x12 x12

( )
24 24x 28
(x 1 2)​​ 2​x​2​1 x 1 5 1 ____
​ ​ ​​ 5 2x3 1 x2 1 5x 1 ____
​ ​​1 4x2 1 2x 1 10 1 ____
​ ​
x12 x12 x12
24x 2 8 24(x 1 2)
5 2x3 1 x2 1 7x 1 10 1 ​______​​5 2x3 1 x2 1 7x 1 10 1 ​________​
x12 x12
5 2x3 1 x2 1 7x 1 10 2 4 5 2x3 1 x2 1 7x 1 6

Thus, the quotient is correct.

6 Sample solution:
a) The result from question 3a) can be written as follows:
23
(8a3 1 22a2 2 21a 2 3) 4 (2a 1 7) 5 4a2 2 3a 1 _____
​ ​
2a 1 7
Performing the reverse operation:

( )

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3 6a 21
(2a 1 7)​​ 4​a​2​2 3a 2 ​_____​ ​ 5 8a3 2 6a2 2 ​_____​1 28a2 2 21a 2 _____
​ ​
2a 1 7 2a 1 7 2a 1 7
6a 1 21 ​
5 8a3 1 22a2 2 21a 2 ​______
​ ​
2a 1 7
3(2a 1 7) ​
5 8a3 1 22a2 2 21a 2 ​_______
​ ​
2a 1 7
5 8a3 1 22a2 2 21a 2 3

Thus, the quotient is correct.

b) The result from 3b) can be written as follows:

7
(3x3 2 6x2 1 10) 4 (x 2 1) 5 3x2 2 3x 2 3 1 ____
​ ​
x21
Replacing variable x by 2:
7
(3(2)3 2 6(2)2 1 10) 4 ((2) 2 1) 5 3(2)2 2 3(2) 2 3 1 ​_____​
(2) 2 1
(24 2 24 1 10) 4 1 5 12 2 6 2 3 1 7
10 5 10

Thus, the quotient is correct.

168 CHAPTER 2 – Algebraic Expressions

 TABLE OF
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ANSWER KEY
CONSOLIDATION 2.2 PAGES 100 TO 103
CHAPTER 23x 1 2) 5 x 1 3x 1 2x 2 3x 2 9x 2 6 5 x 2 7x 2 6
1 a) (x 2 3)(x 1 2 3 2 2 3

b) (2x 1 1)(x2 2 3x 1 3) 5 2x3 2 6x2 1 6x 1 x2 2 3x 1 3 5 2x3 2 5x2 1 3x 1 3

c) (3x 2 2)(4x2 2 x 2 3) 5 12x3 2 3x2 2 9x 2 8x2 1 2x 1 6 5 12x3 2 11x2 2 7x 1 6

d) (2x 1 3)(4x2 2 6x 1 9) 5 8x3 2 12x2 1 18x 1 12x2 2 18x 1 27 5 8x3 1 27

2 a) 4x2 2 8x 2 6 2x 1 1 d) x3 2 3x2 1 0x 1 5 x 1 1
2 (4x2 1 2x) 2x 2 5 2 (x3 1 x2) x2 2 4x 1 4
210x 2 6 24x2 1 0x

2 (210x 2 5) 2 (24x2 2 4x)

21 4x 1 5
2 (4x 1 4)
Therefore you get: 1
(4x2 2 8x 2 6) 4 (2x 1 1) 5 (2x 2 5) 1 _____
21
1
​​ 
2x 1 1
​​ The quotient is x2 2 4x 1 4 1 ______
​ ​​.
or (4x2 2 8x 2 6) 5 (2x 1 1) 3 (2x 2 5) 2 1 x11

b) 9x2 2 15x 1 4 3x 2 4 4 a) The answer is C; that is, the polynomial 2x2 2 3x 2 27.
2 (9x2 2 12x) 3x 2 1

ANSWER KEY
CHAPTER 2
23x 1 4 EXPLANATION:
2 (23x 1 4) Being divisible means that the quotient does not have a
0
remainder. You can check this by carrying out the three
Therefore you get: divisions. However, there is a simpler way.

(9x2 2 15x 1 4) 4 (3x 2 4) 5 3x 2 1 or (9x2 2 15x 1 4) 5 When there is no remainder from the division, the

(3x 2 4) 3 (3x 2 1) polynomial P divided by (x 1 3) gives the polynomial Q;

P
that is, ​​ ____ ​ 5​Q. You could also write P 5 (x 1 3) 3 Q.
x13
3 a) 2x3 2 x2 2 9x 1 11 2x 2 3 This equation is true regardless of the value given to x.
2 (2x3 2 3x2) x2 1 x 2 3 For x 5 23, we find that P 5 0 3 Q, so P 5 0. This needs
2x2 2 9x to be checked for each polynomial: simply replace x with
2 (2x2 2 3x) 23 and note whether the outcome is 0. If not, the
26x 1 11
polynomial is not divisible by x 1 3.
2 (26x 1 9)
(A) By replacing x with 23 in x2 1 9, you get 18 ( 0).
2
The polynomial is not divisible by x 1 3.
2
The quotient is x2 1 x 2 3 1 _______
​ ​​.
2x 2 3 (B) By replacing x with 23 in x2 2 9x 2 27, you get 9 ( 0).

b) The quotient is x2 1 4. The polynomial is not divisible by x 1 3.

(C) By replacing x with 23 in 2x2 2 3x 2 27, you get 0.

x3 2 2x2 1 4x 2 8 x 2 2
The polynomial is divisible by x 1 3.
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2 (x3 2 2x2) x2 1 4
0 1 4x 2 8
2 (4x 2 8) b) 2x2 2 3x 2 27 5 (x 1 3) 3 (2x 2 9)
0
EXPLANATION:
The quotient is x21 4.
By dividing (2x3 2 3x 2 27) by (x 1 3), you get a quotient of
c) x3 1 0x2 1 6x 1 100 x 1 4
(2x 2 9) without a remainder, as expected.
2 (x3 1 4x2) x2 2 4x 1 22
24x2 1 6x 2x2 2 3x 2 27 x 1 3
2 (2x2 1 6x) 2x 2 9
2 (24x2 2 16x) 29x 2 27
22x 1 100
2 (29x 2 27)
2 (22x 1 88) 0
12
12
The quotient is x2 2 4x 1 22 1 ​______​​.
x14

169
 TABLE OF
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ANSWER KEY
5 Box A has the greater capacity.
CHAPTER 2
Sample justification:

Simply calculate the volume of the inside of each box (assuming the thickness of the sides of the box is negligible), then determine
the difference between these volumes.

Volume of Box A: x(x 1 1)(x 1 4) 5 x(x2 1 4x 1 x 1 4) 5 x(x2 1 5x 1 4) 5 x3 1 5x2 1 4x.

Volume of Box B: (x 2 1)(x 1 3)2 5 (x 2 1)(x2 1 6x 1 9) 5 x3 1 6x2 1 9x 2 x2 2 6x 2 9 5 x3 1 5x2 1 3x 2 9.

Difference (first volume minus second): (x3 1 5x2 1 4x) 2 (x3 1 5x2 1 3x 2 9) 5 x 1 9.

Since x has to be positive because it is the height of Box A, x 1 9 is therefore strictly positive. Box A therefore has the greater capacity.

6 Victor is correct because x3 2 8 is divisible by x 2 2. However, he is wrong to say that the quotient is x2 1 4.
The quotient of this division is in fact x2 1 2x 1 4.

To show this, you could calculate the product below; as you can see, all the x2 and x terms cancel each other out.

(x 2 2)(x2 1 2x 1 4) 5 x3 1 2x2 1 4x 2 2x2 2 4x 2 8

5 x3 2 8

7 The area of the large square is c2.

It is made up of four right-angled triangles with the dimensions a 3 b ; the area of each is therefore __
ab
​​  2 ​​ , and so the area for all
four is ___
4ab
​​  2 ​​ 5 2ab.
This book is the property of Dickson Joseph.

The sides of the small square are (a 2 b), therefore, the area is (a 2 b)2 5 a2 2 2ab 1 b2.

Thus c2 5 a2 2 2ab 1 b2 1 2ab 5 a2 1 b2.

8 The First Binomial In this situation, he is right because x is a positive number

greater than 1. Therefore, the square of x is always greater—
324x3 2 45x2 2 99x 2 10 3x 2 5
even much greater—than x. Also, since 36 is greater than
2 (324x3 2 540x2) 108x2 1 165x 1 242
29 in terms of absolute value, the result is always positive.
495x 2 99x
2

2 (495x2 2 825x) Testing the Conjecture

726x 2 10
2 (726x 2 1210) You simply need to find a counterexample.
Remainder 1200
324x3 2 45x2 2 99x 2 10 29x 11
Therefore, 2 (324x3 2 36x2) 236x2 1 x 2 ___
100
​​  9 ​​
(324x3 2 45x2 2 99x 2 10) 4 (3x 2 5) 29x2 2 99x

1200 2 (29x2 1 x)
5 (108x2 1 165x 1 242) 1 _____
​ ​

© SOFAD / All Rights Reserved.

3x 2 5 2100x 2 10

2 (2100x 1 ___
100
In this situation, Hamid is correct because x is a positive ​​  9 )​​
number and all the coefficients are positive; therefore the ​ 2 ___ 190
​  ​​ 9
result will be greater than x.
Hamid’s conjecture is false. If the divisor is 29x 1 1,
The Second Binomial the resulting quotient 236x2 1 x 2 ___
100
​​  9 ​​ is still a number
324x3 2 45x2 2 99x 2 10 9x 1 1 less than x, since the square of x is always greater—even
2 (324x3 1 36x2) 36x2 2 9x 2 10 much greater— than x.
281x2 2 99x

2 (281x2 2 9x)
2 90x 2 10

2 (290x 2 10)
0
(324x3 2 45x2 2 99x 2 10) 4 (9x 1 1) 5 (36x2 2 9x 2 5)

170 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

ANSWER KEY
9 a) 1) For x 5 0, the rational expression is equal
____________
3​(0)​​ ​ 2 2(0) 2 8
2
__
28
d) ​
3​x​2​ 2 2x 2 8
​______________
  
  
(x 2 2)(3x 1 4)
​​5 ______________
​   ​
3​x​2​ 2 10x 1 8 (x 2 2)(3x 2 4)
CHAPTER 2
to ​​    ​​ 5 ​​ 
  
3​(0)​​2​ 2 10(0) 1 8 8
 ​​ 5 21.
4
e) x  2 and x  __
​​
2) For x 5 1, the rational expression is equal 3
to ____________  ​​ 5 _______
​​  3 2 10 1 8 ​​ 5 ___
3​(1)​​2​ 2 2(1) 2 8 32228 27
​​   
   ​​  1 ​​ 5 27. f ) The ratio of the areas is:
3​(1)​​ ​ 2 10(1) 1 8
2

​(x 2 2)​(3x 1 4) 3x 1 4
​​ ______________
   ​​ 5 _______ ​ ​
NOTE: In this problem, the values don’t make sense ​(x 2 2)​(3x 2 4) 3x 2 4
because the rational expression represents the ratio If x 5 5, you get:
between areas, which must be a positive number. 3x 1 4 ________
3(5) 1 4
_______
​ ​​ 5 ​ ​
You can deduce from these results that 0 and 1 are not 3x 2 4 3(5) 2 4
possible values for the variable x. (See next note for
19
more on this subject.) 5 ___
​ ​
11
g) Taking into account the factoring of the two polynomials
b) If x 5 2, the numerator and denominator of the rational
in the numerator and denominator of the original
expression are equal to 0.
expression, you can deduce that Magali has described
As you can see, the dimensions of the rectangular band as follows:
3x2 2 2x 2 8 5 3(2)2 2 2(2) 2 8 5 12 2 4 2 8 5 0 and 3x  4 3x  4
3x2 2 10x 1 8 5 3(2)2 2 10(2) 1 8 5 12 2 20 1 8 5 0.
x2 A B
0
You therefore obtain an indeterminate ratio: __
​ ​​. 6x
0

ANSWER KEY
CHAPTER 2
c) Both polynomials are divisible by (x 2 2).
The common width of the two small rectangles is
In fact, if you divide each polynomial by (x 2 2), (x 2 2) units, while the length of each of the rectangles
the remainder is 0. is (3x 2 4) and (3x 1 4) units; placed end to end, the two
rectangles form a larger rectangle measuring 6x units
(3x2 2 2x 2 8) 4 (x 2 2) 5 (3x 1 4) and
in length.
(3x2 2 10x 1 8) 4 (x 2 2) 5 (3x 2 4)

3x2 2 2x 2 8 x 2 2 NOTE: Given that the length of the banner is (x 2 2) units,

2 (3x2 2 6x) 3x 1 4 the problem wouldn’t make sense if x 5 0 or x 5 1, because
4x 2 8 this would mean that the width of the banner was negative.
2 (4x 2 8) You can also see that, if x 5 2, the banner would have no
0 width. That is why the area of A and B is equal to 0 in this case.

3x2 2 10x 1 8 x 2 2
2 (3x2 2 6x) 3x 2 4
24x 1 8

2 (24x 1 8)
0
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INTEGRATION PAGES 108 TO 111

1 a) 
First, the dividend’s trinomial must be ordered correctly: b) First, the divisor’s binomial must be ordered correctly:

10x2 2 12 1 14x becomes 10x2 1 14x 2 12 4 1 3x becomes 3x 1 4

Since the numerator cannot be factored using a known Next, you can see that the numerator is the difference
special identity, you must proceed with the division: of two squares.

10x2 1 14x 2 12 5x 2 3 9x2 2 16 5 (3x 2 4)(3x 1 4)

2 (10x2 2 6x) 2x 1 4
9​x​2​ 2 16 (3x 2 4)​(3x 1 4)​ 4
20x 2 12 Therefore, ​_________​​5 ______________
​​     ​​, where x  2​__
​  ​​.
2 (20x 2 12) 3x 1 4 ​3x 1 4​ 3
0 4
Simplification gives 3x 2 4, where x  2​__
​  ​​.
3
The quotient is therefore 2x 1 4, if x  22.

171
 TABLE OF
CONTENTS

ANSWER KEY
Validation: Validation:

CHAPTER 2 this with the division algorithm:

You can check You can check this with the division algorithm:

9x2 1 0x 2 16 3x 1 4 4x2 1 0x 2 9 2x 1 3
2 (9x2 1 12x) 3x 2 4 2 (4x2 1 6x) 2x 2 3
212x 2 16 26x 2 9

2 (212x 2 16) 2 (26x 2 9)

0 0
4
The quotient is therefore 3x 2 4, where x  2​__
​  ​​. d) First, the dividend’s trinomial must be ordered correctly:
3
c) First, the dividend’s binomial must be ordered correctly: 10x2 2 16 2 12x becomes 10x2 2 12x 2 16
29 1 4x2 becomes 4x2 2 9 Since the numerator can’t be factored using a known
special identity, you must proceed with the division:
Next, notice that the numerator is a difference of
two squares. 10x2 2 12x 2 16 5x 1 4
2 (10x2 1 8x) 2x 2 4
4x2 2 95 (2x 2 3)(2x 1 3) 2 20x 2 16
2 (220x 2 16)
4​x​2​ 2 9 (2x 2 3)​(2x 1 3)​ 3
Therefore, ​________​​5 ______________
​​     ​​, where x  2​__
​  ​​. 0
2x 1 3 ​2x 1 3​ 2
4
3 The quotient is therefore 2x 2 4, where x  2​__
​  ​​.
Simplification gives 2x 2 3, where x  2​__
​  ​​. 5
2

2 a) 1) (a 1 b)3 5 (a 1 b)(a 1 b)2 5 (a 1 b)(a2 1 2ab 1 b2) 5 a3 1 2a2b 1 ab2 1 a2b 1 2ab2 1 b3 5 a3 1 3a2b 1 3ab2 1 b3
This book is the property of Dickson Joseph.

2) (a 1 b)4 5 (a 1 b)(a 1 b)3 5 (a 1 b)(a3 1 3a2b 1 3ab2 1 b3) 5 a4 1 3a3b 1 3a2b2 1 ab3 1 a3b 1 3a2b2 1 3ab3 1 b4
5 a4 1 4a3b 1 6a2b2 1 4ab3 1 b4

b) (a 1 b)5 5 a5 1 5a4b 1 10a3b2 1 10a2b3 1 5ab4 1 b5

EXPLANATION:
The triangle below can be used to determine the coefficients. For the exponents of variables, simply order them from 5 to 0 for
the variable a and from 0 to 5 for the variable b. (Each term of the polynomial is to the fifth degree.)

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1

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3 Sample proof:
Two consecutive odd numbers can be represented by the expressions 2n 1 1 and 2n 1 3.

The difference of their squares is (2n 1 3)2 2 (2n 1 1)2. This expression can be simplified.

(2n 1 3)2 2 (2n 1 1)2 5 (4n2 1 12n 1 9) 2 (4n2 1 4n 1 1) 5 8n 1 8 5 8(n 1 1)

Since n is an integer, n 1 1 is also an integer. The result is therefore 8 times an integer. In other words, it is a multiple of 8.

172 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

ANSWER KEY
4 a) Each person will get at least (2n 2 1) pieces; 6 Sample solution:
CHAPTER 2
only 1 piece will be assigned randomly.
Let x equal the length of one side of Matilda’s room.
Sample solution: The dimensions of Ethan’s room are (x 1 1) m and (x 2 1) m.
Number of pieces of chocolate: n(2n 1 1) 5 2n2 1 n. Area of Matilda’s room: x2.
2n2 1 n must be divided by n 1 1. The outcome Area of Ethan’s room: (x 1 1)(x 2 1) 5 x2 2 1.
is a quotient of (2n 2 1) with a remainder of 1.
Matilda’s room is bigger (1 m2 more than Ethan’s).
Symbolically, you have the following equation:
(2n2 1 n) 5 (n 1 1)(2n 2 1) 1 1.
NOTE: The perimeter of the two rooms is the same. However,
b) Sample validation: in the context of the size of rooms, it’s the floor area that must
be compared.
1) The quotient can be checked by performing the
reverse operation:
x
( ​
1
(n 1 1)​​ 2n 2 1 1 ____
n11 )
​​
x1
n 1
5 2n2 2 n 1 ​____​​1 2n 2 1 1 ____
​ ​ x
n11 n11
n 11 x1
5 2n2 1 n 2 1 1 ​____​
n11
5 2n2 1 n 2 1 1 1

ANSWER KEY
CHAPTER 2
7 A) Since the area of the large square as a whole is a2 and
5 2n2 1 n.
the area that has been removed is b2, the area of the
The result is 2n 1 n, which is the total number of
2
piece of cardboard is a2 2 b2.
pieces of chocolate.
b) The width is (a 2 b) and the length is (a 1 b).
2) You could also redo the problem for a particular value c) The special identity is the difference of two squares:
of n; for example, n 5 4. In this case, a chocolate bar a2 2 b2 5 (a 1 b)(a 2 b).
must be shared between 5 people (that is, 4 1 1). The
bar contains 4 rows of 9 pieces (that is, 2 3 4 1 1).
JUSTIFICATION : By cutting the cardboard into two
There are 36 pieces in total; that is, 4 3 9. Each person pieces and sticking them back together, the area does not
should therefore get 7 pieces (which corresponds to change. Therefore, the area of the large rectangle in the third
image is equal to that of the initial shape.
the quotient (2n 2 1) and there will be 1 piece left to
assign randomly.
8 a) Sample answer: The denominators must be two
3) The division can also be checked using the
consecutive integers.
reverse operation; that is, by calculating
1 1 ___ 5 4 1 1 1
(n 1 1)(2n 2 1) 1 1. Example: __
​ ​2 __
​ ​​5 ​ ​​2 ___
​ ​​5 ​___​​5 __
​ ​​3 __
​​
4 5 20 20 20 4 5
(n 1 1)(2n 2 1) 1 1 1 1 1 1
​__​​2 ______
b) ​ ​ ​​5 __
​ ​​3 ______
​ ​​
5 (2n2 2 n 1 2n 2 1) 1 1 n n11 n n11
© SOFAD / All Rights Reserved.

5 2n2 1 n
c) Sample proof:
The answer is the dividend.
To prove that it is an identity, you can simplify each side
of the equation and see whether you obtain the same
5 a) Since the two binomials are squared, you get perfect
expression.
square trinomials:
Left-hand side of the equation:
(9x 2 1)2 2 (7x 2 1)2 5 (81x2 2 18x 1 1) 2 (49x2 2 14x 1 1)
1
__ 1
5 32x2 2 4x ​ ​​2 ______
​ ​
n n11
The area of the path is represented by the algebraic n11 n
5 ​________​​2 ________
​ ​
expression 32x2 2 4x. n(n 1 1) n(n 1 1)
(n 1 1) 2 n
b) (9x 2 1)2 2 (7x 2 1)2 5 ​__________​
n(n 1 1)
5 ((9x 2 1) 1 (7x 2 1))((9x 2 1) 2 (7x 2 1))
1
5 (16x 2 2)(2x) 5 32x2 2 4x 5 ​________​​.
n(n 1 1)
Right-hand side of the equation:
1
__ 1 1
​ ​​3 ______
​ ​​5 ________
​ ​​.
n n 1 1 n(n 1 1)

173
 TABLE OF
CONTENTS

ANSWER KEY
9 a) To determine the number of boxes that can be stored in the warehouse, you must divide each dimension of the warehouse by
CHAPTER 2
the length of one of the sides of a box, then multiply the three answers.

Lengthwise: (15x2 2 34x 1 8) 4 (x 2 2) 5 (15x 2 4) It holds 15x 2 4 boxes lengthwise.

Widthwise: (8x 2 12x 2 8) 4 (x 2 2) 5 (8x 1 4)

2
It holds 8x 1 4 boxes widthwise.

Heightwise: (6x2 2 8x 2 7.5) 4 (x 2 2) 5 (6x 1 4) 1 ____ 0.5

​​  x 2 2 ​​ It holds 6x 1 4 boxes heightwise, with a remainder of ____
0.5
​​  x 2 2 ​​ box.
Keeping the integer part only, that is 6x 1 4.

Therefore, the warehouse can hold a maximum of (15x 2 4)(8x 1 4)(6x 1 4) boxes.

(15x 2 4)(8x 1 4)(6x 1 4) 5 (120x2 1 28x 2 16)(6x 1 4)

5 720x3 1 648x2 116x 2 64

b) It is not possible to completely fill the warehouse with boxes, because there is 0.5 metre of height remaining, which represents
a fraction of a box of ____
0.5
​​  x 2 2 ​​ .

To calculate the empty space, multiply the leftover height by the length and width.

14(15x2 2 34x 1 8)(8x2 2 12x 2 8)

5 14(120x4 2 180x3 2 120x2 2 272x3 1 408x2 1 272x 1 64x2 2 96x 2 64)
5 14(120x4 2 452x3 1 352x2 1 176x 2 64)
5 14(120x4 2 452x3 1 352x2 1 176x 2 64)
5 56(30x4 2 113x3 1 88x2 1 44x 2 16) or 1680x4 2 6328x3 1 4928x2 1 2464x 2 896

So there is 1680x4 2 6328x3 1 4928x2 1 2464x 2 896 m3 of empty space.

This book is the property of Dickson Joseph.

10 a) For this expression to be properly defined, it is essential that x  0 and x  21.

EXPLANATION:
It is clear that the value of x cannot be 21; otherwise, the denominator of the rational expression ____
1
​​  ​​ would be equal to 0.
x11
In addition, ____ ​​ cannot equal 1, since the denominator of the complex rational expression, that is, 1 2 ____
1 1
​​  ​​  ​​ , would be 1 2 1, which is 0.
x11 x11
As a consequence of this condition, x cannot equal 0. In effect:
1
​______
​ ​ 1​
x11
1x11
x0

b) When x 5 1, When x 5 2, When x 5 3,

(1) 1 (3) 3
​​ ___________  ​​ 5 ______
​​   ​​ (2) 2 ​​ ___________  ​​ 5 ______
​​   ​​
1 2 ​ 
1
_______ ​ 1 2 __
1
​  2 ​ ​​ ___________1
 ​​ 5 ______
​​  1
 ​​ 1 2 ​ 
1
_______ ​ 1 2 __
1
​  4 ​
((1) 1 1) 1 2 _______
​   ​ 1 2 __ ​  3 ​ ((3) 1 1)
1 ((2) 1 1) 3
5 __
​​  1  ​​ 2 5 __
​​  3  ​​
5 __

© SOFAD / All Rights Reserved.

__
​  2 ​ ​​  2  ​​ __
​  4 ​
__
​  ​
2 3 4
5 1 3 __
​​ 3 5 3 3 __
​​
1 5 2 3 __
​​ 3
52 2 54
53

You can see that the result is always 1 more than the value of x.

c) The expression must be simplified, which can be done as follows:

x
_________ x x x x x 1 1 ________
x​(x 1 1)​
​​   ​​ 5 ____________
​​      ​​ 5 _________
​​   ​​ 5 _____
​​  x  ​​ 5 x 4 ______
​ ​​5 x 3 ______
​ ​​5 ​ ​​5 x 1 1.
1 x11 1 (x 1 1) 2 1 _____
1 2 _____
​  x 1 1 ​ _____ ​  x 1 1 ​ 2 _____
​  x 1 1 ​ _________
​  ​ ​  x 1 1 ​ x11 x x
x11

a
__ c a3d2c3b
​ ​​2 __
11 a) ​ ​ ​​5 ________________
​   ​
b d b3d
a c _______
__ a3c
​ ​​3 __
b) ​ ​ ​​5 ​ ​
b d b3d
a c _______
__ a3d
​ ​​4 __
c) ​ ​ ​​5 ​ ​
b d b3c

174 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

ANSWER KEY
LES
CHAPTER 2 Mentalists PAGES 112 AND 113
The Amateur
Checking the Friend’s Assertion

To check the assertion of the students’ friend, you must first translate Monica’s method into an algebraic expression. You must then
simplify the algebraic expressions of Monica and Armando.

For Monica’s Method

Monica’s method is translated into the following algebraic expression:

3​x​2​2 3
__________
​ ​
(6x 1 3)(x 1 1)

The expression can then be simplified to get an equivalent algebraic expression:

3(​x​2​2 1) 3(x 2 1)(x 1 1) (x 2 1)
5 ​___________​​5 ​_____________​​5 ______
​ ​
3(2x 1 1)(x 1 1) 3(2x 1 1)(x 1 1) (2x 1 1)

For Armando’s Method

The problem can be broken down into three steps.

6​x​3​1 3​x​2​1 5x 1 1
_____________

ANSWER KEY
CHAPTER 2
​ ​​1 (​ 2​x​2​2 3x 1 2)(3​x​2​1 2x 2 1)​2 ​x​ ​2​(3x 2 1)(2x 2 1)​2 7x
2x 1 1
To make this algebraic expression easier to simplify, it will be split into four parts:

Step 1: Product of the trinomials

(2x 2 2 3x 1 2)(3x 2 1 2x 2 1)
5 6x4 1 4x3 2 2x2 2 9x3 2 6x2 1 3x 16x2 1 4x 2 2
5 6x4 2 5x3 2 2x2 1 7x 2 2

Therefore, this part of the expression is equivalent to 6x4 2 5x3 2 2x2 1 7x 2 2.

Step 2: Product of three factors

x 2(3x 2 1)(2x 2 1)
5 x2(6x2 2 3x 2 2x 1 1)
5 x2(6x2 2 5x 1 1)
5 6x4 2 5x3 1 x2

Therefore, this part of the expression is equivalent to 6x4 2 5x3 1 x2.

Step 3: Simplification
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Finally, putting the parts back together (not forgetting the monomial 27x at the end):

6​x​3​1 3​x​2​1 5x 1 1
​_____________
   ​​1 (6x4 2 5x3 2 2x2 1 7x 2 2) 2 (6x4 2 5x3 1 x2) 2 7x
2x 1 1
6​x​3​1 3​x​2​1 5x 1 1
5 _____________
​   ​​1 6x4 2 5x3 2 2x2 1 7x 2 2 2 6x4 1 5x3 2 x2 2 7x
2x 1 1
6​x​3​1 3​x​2​1 5x 1 1
5 ​_____________
   ​​2 3x2 2 2
2x 1 1
6​x​3​1 3​x​2​1 5x 1 1 ______________
(23​x​2​2 2)(2x 1 1)
5 _____________
​ ​​1 ​   ​
2x 1 1 2x 1 1
6​x​3​1 3​x​2​1 5x 1 1 _______________
(26​x​3​2 3​x​2​2 4x 2 2)
5 _____________
​ ​​1 ​   ​
2x 1 1 2x 1 1
x21
5 _____
​ ​
2x 1 1

175
 TABLE OF
CONTENTS

ANSWER KEY
Since each method is equivalent to the algebraic expression ______
(x 2 1)
​​  (2x 1 1) ,​​ their friend is right. However, although the two methods are

CHAPTER 2
mathematically equivalent, Monica’s is less complex and more convenient for audience members.

Interpretation of the Final Answers

Regardless of the number picked by the audience member, Armando and Monica know that the answer will be a fraction whose
numerator will be 1 less than the chosen value and the denominator will be 1 more than the double of the chosen value.

The final answers will be:

Number
Value as a fraction
chosen
((1) 2 1) 0
1 ​_________​5 __
​ ​5 0
(2(1) 1 1) 3
((2) 2 1)
_______ 1
2 ​ ​5 __
​ ​
(2(2) 1 1) 5
((3) 2 1) 2
3 ​_______​5 __
​ ​
(2(3) 1 1) 7
((4) 2 1) 3 1
4 ​_______​5 __
​ ​​5 __
​​   ​​
(2(4) 1 1) 9 3
((5) 2 1)
_______ 4
5 ​ ​5 ___
​ ​
(2(5) 1 1) 11
((6) 2 1)
_______ 5
6 ​ ​5 ___
​ ​
(2(6) 1 1) 13
((7) 2 1)
_______ 6 2
7 ​ ​5 ___
​ ​​5 __
​​   ​​
(2(7) 1 1) 15 5
This book is the property of Dickson Joseph.

((8) 2 1) 7
8 ​_______​5 ___
​ ​
(2(8) 1 1) 17
((9) 2 1)
_______ 8
9 ​ ​5 ___
​ ​
(2(9) 1 1) 19

You may now fill in the self-evaluation grid for the five targeted criteria. Refer to the grid at the end of the guide.
Your teacher or tutor may also provide you with the evaluation indicators for this LES. These will help you judge the
quality of your problem-solving skills.

© SOFAD / All Rights Reserved.

176 CHAPTER 2 – Algebraic Expressions

 TABLE OF
CONTENTS

REFRESHER
CHAPTER 1 PAGES 115 TO 120
1 a) Independent variable: the number of words in a text. b) Amount of water in the glass
Dependent variable: the time it takes to write them. as a function of time
Amount
(ml)
b) Independent variable: the surface area to be painted. 360
Dependent variable: the number of paint cans to be
bought. 300 120


c) Independent variable: the average speed of a vehicle. 240

 1
Dependent variable: the time it takes to make the
180 120
journey. 

120
2 Graphs A and B do not represent a function.  1
60  120
Sample justification:

In order to be a function, any value of the independent  1 1

0 2 3 4 5
variable (x) may be associated with only one value of the Time
dependent variable (y). Visually, this means that if you were to (s)
draw an imaginary vertical line on the graph, it would touch
c) 240 ml
the curve of the function at only one point. This is clearly not
the case for A, bacause the graph itself is a vertical line. This is
EXPLANATION: The range can be found by looking at the
not the case for B either, as the illustration below shows. graph and seeing that it goes through the point (2, 240).

y It can also be calculated as follows:

f(2) 5 120(2) 5 240 ml.
4
2 () 1
d) f ​​ __ () 1
​ ​ ​​ 5 120​​ __
2
​ ​ ​5 60 ml
2
0
f(1.5) 5 120(1.5) 5 180 ml

REFRESHER
ANSWER KEY
24 2
2 2 4 x
2
2
e) The time it takes to fill the glass varies from 0 to 3 s,
4
2
so the domain is [0, 3] s.

f ) The amount of water in the glass varies from 0 to 360 ml,

3 a) f(t) 5 120t
so the range is [0, 360] ml.
Sample solution:
g) 1) Sample solution:
The ratio _​​  x ​​ 5 ___
​​  1 ​​ 5 ___
y 120 360
​​  3 ​​ 5 120 is constant. This is therefore
150 5 120t
a directly proportional relationship of the form y 5 ax,
150
where a is the coefficient of proportionality. In this case, t 5 ____
​ ​​5 1.25
120
a 5 120: the tap pours 120 ml of water per second.
© SOFAD / All Rights Reserved.

The glass will contain 150 ml after 1.25 s.

2) Sample solution:

If you double the amount of water (150 3 2 5 300 ml),

you double the time (1.25 3 2 5 2.5 s).

The glass will contain 300 ml after 2.5 s.

4 a) f(2) 5 4(2) 2 6 5 2
b) f(1.5) 5 4(1.5) 2 6 5 0

c) f(23) 5 4(23) 2 6 5 218

d) f(0) 5 4(0) 2 6 5 26

177
 TABLE OF
CONTENTS

ANSWER KEY
5 Interval notation Set-builder notation Number line

REFRESHER
a) [2, 5[ {x  r  2  x , 5}
3
2 2 2 2 1 0 1 2 3 4 5 R

b) ]0, 5] {x  r | 0 , x  5}
3
2 2 2 2 1 0 1 2 3 4 5 R

c) ]22, 3] {x  r  22 , x  3}
3
2 2 2 2 1 0 1 2 3 4 5 R

d) ]21, 3[ {x  r  21 , x , 3}
3
2 2 2 2 1 0 1 2 3 4 5 R

e) ]2∞, 0[ {x  r  x , 0} 2 3 2 2 2 1 0 1 2 3 4 5 R

f) [1, 1∞[ {x  r  x  1}
3
2 2 2 2 1 0 1 2 3 4 5 R

g) ]2∞, 2] {x  r  x  2} 2 3 2 2 2 1 0 1 2 3 4 5 R

h) ]2, 1∞[ {x  r  x . 2}
3
2 2 2 2 1 0 1 2 3 4 5 R

6 Algebraic Interval 9 a) Strictly negative

Number line
This book is the property of Dickson Joseph.

representation notation
b) Negative
a) x  8 [8, 1∞[
8 R c) Strictly positive

b) x , 2 ]2∞, 2[
d) Positive
2 R

10 a) Strictly positive: ]22, 6[

c) 22  x , 3 [22, 3[
2
2 3 R Strictly negative: ]2∞, 22[  ]6, 1∞[

1 1
]2∞, 2​__
b) Positive: ]2∞, 2]
d) x  2​__
​  ​​ __1 ​  ​​]
2 R 2 Strictly negative: ]2, 1∞[
2
2

e) 23 ,x,
]23, 0[ c) Strictly positive: ]2∞, 2[  ]2, 1∞[
0 3 0 R Negative: [22, 2]
2

f) x  4 or ]2∞, 4]  d) Strictly positive: ℝ

x . 10 4 10 R ]10, 1∞[
Strictly negative: 
g) x  2 or ]2∞, 2] 

© SOFAD / All Rights Reserved.

x.3 2 3 R ]3, 1∞[ 11 a) 10 b) 0

h) 4 , x , 6 ]4, 6[ c) 8 d) 1
4 6 R
e) 967 f ) 1280
7 a) [5, 1∞[
b) [2, 6]

c) {x  r  x . 2}

d) {x  r  25  x  4}

8 C and D are included in the intervals provided.

178 ANSWER KEY

 TABLE OF
CONTENTS

ANSWER KEY
CHAPTER 2 PAGES 127 TO 130
REFRESHER 1
12 It is missing ___
​ ​​.
64
In effect,
__ 1 1 __ 1 1 1 1 32 16 ___ 8 4 2 1 63
​​   ​​ 1 __
​​   ​​ 1 ​​   ​​ 1 ___
​​   ​​ 1 ___
​​   ​​ 1 ___
​​   ​​ 5 ___
​​   ​​ 1 ___
​​   ​​ 1 ​​   ​​ 1 ___
​​   ​​ 1 ___
​​   ​​ 1 ___
​​   ​​ 5 ___
​​   ​​
2 4 8 16 32 64 64 64 64 64 64 64 64
63 1
and 1 2 ___ ​​   ​​ 5 ___ ​​   ​​.
64 64

__ 1 1 5 1 6 3 12 4 3 4
​ ​​1 ___
13 a) ​ ​ ​​5 ___
​ ​​1 ___
​ ​​5 ___
​ ​​5 ___
​ ​ ______
15 a) ​
​ ​​5 __
​​
4 20 20 20 20 10 15 4 3 5

1
__ 1 5 1 4 1 50 4 10 __ 5
​ ​​2 ___
b) ​ ​ ​​5 ___
​ ​​2 ___
​ ​​5 ___
​ ​​5 __
​​ _______
b) ​
​ ​​5 ​ ​
4 20 20 20 20 5 60 4 10 6

1
__ 1 1 14 4 14 __ 1
​ ​​3 ___
c) ​ ​ ​​5 ___
​ ​ _______
c) ​
​ ​​5 ​ ​
4 20 80 84 4 14 6

1
__ 1 1 20 ___ 20 48 4 6 ___ 8
​ ​​4 ___
d) ​ ​ ​​5 __
​ ​​3 ___
​ ​​5 ​ ​​5 5 ______
d) ​
​ ​​5 ​ ​
4 20 4 1 4 90 4 6 15

5
__ 8 5 3 5 ______8 3 2 ___25 16 ___ 41 44 4 4 ___ 11
​ ​​1 ___
e) ​ ​ ​​5 _____
​ ​​1 ​ ​​5​​ ​1 ___
​ ​​5 ​ ​ ______
e) ​
​ ​​5 ​ ​
6 15 6 3 5 15 3 2 30 30 30 28 4 4 7

5
__ 8 25 16 ___ 9 3 33 4 11 __ 3
​ ​​2 ​___
f ) ​ ​ ​​5 ___
​ ​​2 ___
​ ​​5 ​ ​​5 ___
​ ​ _______
f ) ​
​ ​​5 ​ ​
6 15 30 30 30 10 55 4 11 5

5
__ 8 40 4 84 4 21 4
​ ​​3 ___
g) ​ ​ ​​5 ___
​ ​​5 __
​​ ________
g) ​
​ ​​5 __
​​
6 15 90 9 105 4 21 5

5
__ 8 5 15 ___ 75 25 81 4 3 ___ 27
​ ​​4 ___
h) ​ ​ ​​5 __
​ ​​3 ___
​ ​​5 ​ ​​5 ___
​ ​ ______
h) ​
​ ​​5 ​ ​
6 15 6 8 48 16 21 4 3 7

REFRESHER
ANSWER KEY
105 4 3 ___
_______ 35
14 a) Divisors of 15: 1, 3, 5, 15 i) ​
​ ​​5 ​ ​
267 4 3 89
Divisors of 90: 1, 2, 3, 6, 9, 10, 15, 30, 45, 90
15 4 15 __
_______ 1 60 4 12
________ 5
​ ​​5 ​ ​ j) ​
​ ​​5 ___
​ ​
90 4 15 6 168 4 12 14

b) Divisors of 28: 1, 2, 4, 7, 14, 28

16 a) 4 3 1.5 3 x1 1 1 5 6x2
Divisors of 63: 1, 3, 7, 9, 21, 63
28 4 7 __
______ 4 b) 4x2 1 1 5 4x3
​ ​​5 ​ ​
63 4 7 9
c) (3x3)(3x3) 5 9x3 1 3 5 9x6
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c) Divisors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

d) 4 3 3 3 x2 3 y 5 12x2y
Divisors of 16: 1, 2, 4, 8, 16
1
72 4 8 __
______ 9 e) 6 3 __
​ ​​3 x 3 y1 1 1 5 3xy2
​ ​​5 ​ ​ 2
16 4 8 2
f ) 22 3 21 3 x2 1 1 3 y2 5 2x3y2
d) Divisors of 8: 1, 2, 4, 8
Divisors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 __ 6
17 a) ​
​ ​x1 2 1 5 3
844
______ 2 2
​ ​​5 __
​​
36 4 4 9 8
__ 8
​ ​x3 2 1 5 __
b) ​ ​ ​x2 5 1.6x2
5 5
e) Divisors of 32: 1, 2, 4, 8, 16, 32
10​x​1 2 2​
______ 2
Divisors of 18: 1, 2, 9, 18 c) ​
​ ​​5 2x 21 5 __
​​
5 x
32 4 2 ___
______ 16
​ ​​5 ​ ​ 3​x​1 2 1​y
______
18 4 2 9 d) ​
​ ​​5 y
3
​x​1 2 1​y​2 2 1​ __
_______ y
e) ​
​ ​​5 ​ ​​5 0.5y
2 2
4​x​2 2 1​y
______
f ) ​
​ 2 ​​5 22xy
2

179
 TABLE OF
CONTENTS

ANSWER KEY
18 a) (x 1 6)(2x 21) 5 (x)(2x) 1 (x)(21) 1 (6)(2x) 1 (6)(21) 5 2x2 111x 2 6
REFRESHER
b) (5x 1 3)(3y 1 5) 5 (5x)(3y) 1 (5x)(5) 1 (3)(3y) 1 (3)(5) 5 15xy 1 25x 1 9y 115

c) (6x 2 5)(6x 1 5) 5 (6x)(6x) 1 (6x)(5) 1 (25)(6x) 1 (25)(5) 5 36x2 2 25

19 a) (2x 2 5)(x 2 3) 5 2x2 2 6x 2 5x 1 15 5 2x2 2 11x 1 15

b) (x 1 6)(3x 1 2) 5 3x2 1 2x 1 18x 1 12 5 3x2 1 20x 1 12

c) (3x 2 4)(x 1 8) 5 3x2 1 24x 2 4x 2 32 5 3x2 1 20x 2 32

d) (5x 1 3)(5x 2 3) 5 25x2 2 15x 1 15x 2 9 5 25x2 2 9

20 a) 4x(2x 1 3)
b) 2x(3y 2 4)

c) 5x2(2x 1 1)

d) 3xy(4x 2 7y)

e) 3x2(5x 2 10a 1 3b)

21 a) 78.56 1 (49.967 2 1.1) 5 78.56 1 48.867 5 127.427

b) 2.86(23.8) 5 210.868
This book is the property of Dickson Joseph.

c) 34.2 1 (6.7 2 5.1) 2 (26) 5 34.2 1 1.6 1 6 5 41.8

d) 1.1 2 (4.5 1 2.3(3.4 2 2.6)) 5 1.1 2 (4.5 1 2.3(0.8)) 5 1.1 2 (4.5 1 1.84) 5 1.1 2 6.34 5 25.24

e) 2.1 4 2.1 2 2.1(2.1 2 2.1(2.1 2 2.1) 2 2.1) 5 2.1 4 2.1 2 2.1(2.1 2 0 2 2.1) 5 1 2 2.1(0) 51 2 0 5 1

22 a) (0.0075) 3 3 2 4(23) 5 0.0225 1 12 5 12.0225

b) 100 2 3(210) 5 130

c) 2000 1 4800 5 6800

d) 0.01((150)(24) 1 100 2 20(10)) 5 0.01(2600 1 100 2 200) 5 27

23 a) 1 26 a) 2952 24
2 24 123
b) 2 55
2 48
c) 3
72

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d) 0 2 72
0
24 The degree of a polynomial is the highest degree of the
monomials of which it is the sum. b) 5525 17
2 51 325
Example: The polynomial 5x2 2 4xy 1 3xy2 is the sum of the 42
monomials 5x2 (2nd degree), 24xy (2nd degree) and 3xy2 2 34
(3rd degree). 85
2 85
The degree of this polynomial is therefore 3. 0

25 a) 3x 2 1 c) 5024 32
2 32 157
b) 2y 2 0.5x 182
2 160
c) 3x 2 2
224
d) 22x2 2 4x 1 1 2 224
0

180 ANSWER KEY

 TABLE OF
CONTENTS

EVALUATION GRID
Competency 1: Uses strategies to solve situational problems

Evaluation Excellent Very good Good Poor Very poor

criteria A B C D E

1.1 Identifies Identifies nearly Identifies Identifies Identifies very

Indication of all relevant all relevant some relevant little relevant little relevant
an appropriate information. information. information. information. information.
understanding of the
situational problem

1.2 Uses all relevant Uses nearly Uses some Uses few Uses no
Application of strategies. all relevant relevant relevant relevant
strategies and strategies. strategies. strategies or strategies or
knowledge* does so with does so with
appropriate to the difficulty. great difficulty.
situational problem.

* The evaluation pertains to the strategies applied.

Competency 2: Uses mathematical reasoning

Evaluation Excellent Very good Good Poor Very poor

criteria A B C D E

2.1 Uses all Uses nearly Uses some Uses necessary Uses necessary
Correct use of necessary all necessary necessary mathematical mathematical
appropriate mathematical mathematical mathematical knowledge knowledge
mathematical knowledge and knowledge and knowledge and with difficulty with great
concepts and obtains all the obtains nearly obtains some and obtains few difficulty and
processes correct results all the correct of the correct of the correct obtains very
results. results. results. few of the
correct results.

EVALUATION
2.2 Presents an Presents an Presents an Presents an Presents an
© SOFAD / All Rights Reserved.

GRID
Proper approach that is approach that approach approach that approach that
implementation consistent with is consistent that is fairly is lacking in is very lacking
of mathematical all the selected with nearly all consistent with consistency. in consistency.
reasoning suited to strategies and the selected the selected
the situation. knowledge. strategies and strategies and
knowledge. knowledge.
2.3 Presents an Presents an Presents an Presents an Presents an
Proper organization approach that approach approach that is approach that approach
of the steps in is complete and that is fairly fairly complete is incomplete that is very
an appropriate well organized complete and but not well and not well incomplete and
procedure. and adheres to well organized organized and organized and disorganized
all mathematical and adheres adheres to some adheres to few and adheres
conventions. to nearly all mathematical mathematical to very few
mathematical conventions. conventions. mathematical
conventions. conventions.

181
This book is the property of Dickson Joseph.
 The SOLUTIONS series covers all the courses in the Diversified
Basic Education Program, including the Secondary IV Cultural,
Social and Technical (CST) and Science (Sci) options.
RÉSOLUTION RÉSOLUTION
RÉSOLUTION
RÉSOLUTION
RÉSOLUTION
ES The SOLUTIONS learning approach is based on
UM
LEARNING GUIDE LEARNING GUIDE
MATHEMATICS DBE
VO
L LEARNING GUIDE
MATHEMATICS
MATHEMATICS DBE
DBE
the acquisition of all the prescribed mathematical
2
knowledge in a problem-solving context. The learning
MTH-4151-1 CST MTH-4171-2
MTH-4153-2 SCI sequence that supports this approach is as follows:
ALGEBRAIC AND GRAPHICAL MODELLING

MTH-4271-2
MTH-4253-2
VOLUME 1 SCI
ALGEBRAIC AND
IN A FUNDAMENTAL CONTEXT 1

GRAPHICAL MODELLING ALGEBRAICVOLUME

AND 2
IN A GENERAL CONTEXT 1 ALGEBRAIC
GRAPHICAL AND
MODELLING
IN AGRAPHICAL MODELLING
FUNDAMENTAL CONTEXT 1

PRESENTATION OF A
IN A FUNDAMENTAL CONTEXT 1

SITUATIONAL PROBLEM

EXPLORATION
AN CE
PL I E W
AN CE
PL I E W
OF PROBLEM
C O M TH E N COM E N
IN H
RAM IN H TH
RAM
AN CE

P R O GS T U DY P RWOITGS TGURDAY M
WIT WIT C O M PL IE NEW
IN H TH

OF O FP R O S T U DY
OF KNOWLEDGE
ACQUISITION

RÉSOLUTION RÉSOLUTION PROBLEM-SOLVING

This book is the property of Dickson Joseph.

CONSOLIDATION
LEARNING GUIDE LEARNING GUIDE
MATHEMATICS DBE MATHEMATICS DBE
OF LEARNING

MTH-4152-1 CST MTH-4272-2

MAT-4253-2 SCI
DATA COLLECTION
IN A GENERAL CONTEXT
DATA
COLLECTION
Inductive and deductive questions give meaning
IN A FUNDAMENTAL CONTEXT
to the knowledge and strategies to be acquired.
The learning guides offer a multitude of simple
exercises and more complex tasks to meet the needs
expressed by learners and teachers. Additional
M PL N
CE
I AN E W
M PL
CE
I AN E W
N
resources are available on the online portal.
RAM RAM
C O TH E C O TH E
IN H IN H
P R O GS T U DY P R O GS T U DY
WIT WIT

OF OF

Components of the SOLUTIONS series:

RÉSOLUTION RÉSOLUTION • Learning guide: print and PDF versions;
• Teaching guide (PDF);
LEARNING GUIDE LEARNING GUIDE
MATHEMATICS DBE MATHEMATICS DBE • Videos on situational problems;
• ICT Activities: GeoGebra, graphing calculator;
MTH-4153-2 CST MTH-4273-2
MAT-4253-2 SCI
GEOMETRIC GEOMETRIC • Scored activities;
REPRESENTATION REPRESENTATION
IN A GENERAL CONTEXT IN FUNDAMENTAL CONTEXT 1

• Answer keys.

CE CE
I AN E W I AN E W
M PL N M PL N
RAM
C O TH E
RAM
IN C O TH E
H IN H
P R O GS T U DY P R O GS T U DY
WIT WIT

OF OF

8816-01