Guided activities for precalculus

Aligned to Precalculus by Stitz and Zeager

Dr. Philip DeOrsey

Westfield State University
pdeorsey@westfield.ma.edu

Version 1
Last updated: January 12, 2019
 Notes to the teacher

These activities are designed as a guide to help your students navigate standard precalculus content.
The purpose is to shift the focus from you, the instructor, to the student(s) working in small groups. When
I teach precalculus a typical class period begins with me briefly introducing the daily topic, and setting
the student expectations for that class period. Then students will work in small groups through the guided
activities while I move around the room helping them. If I find there is a lot of struggle surrounding a
particular topic I typically bring everyone together to discuss. Ideally there will be a student who can share
their progress but if no students are able to find success I will go to the board and work through some
examples.
As you read through the materials you will notice that there are places where students are directed to
come back together for an in class discussion. These questions often serve as reminders to me to bring the
class back together and see what everyone has discovered, even if everyone is doing well. The hope is to
have students discuss the content, and I will ask individual students to share what they have found. This
gives me a little more control over which discoveries are emphasized, and I feel it brings a more uniform
understanding amongst the students.
It is common in guided activities to have space for students to write their answers directly on the guides,
however I chose to intentionally leave very little space for students to write on the sheets. My intention is
to get students to keep a notebook as a companion to working through these materials. I have observed
students often get a lot from rephrasing the questions and solution in their own words and this is my effort
to help them do that.
Each section in these materials begins with a book reference. This is the section in “Precalculus” by Stitz
and Zeager that students should reference if they want additional material. I chose to align my materials
with the book by Stitz and Zeager for a few reasons: it is freely available online to all students, it has
hundreds of questions on each topic along with their answers, and it covers everything I want and much
more. I often direct students to the questions at the end of each section for additional practice.
For many of the activities I have created Desmos graphs to complement the student work. If you have
enough time I feel it is valuable to have students create many of these graphs, but if you are in a time crunch
they are available here for you to use.
To go along with each section I have created a video that serves as an additional resource for students.
I have found that many students find comfort in having a lecture, or video, or some other explanation to
listen to and these are my way of filling that need. Using these videos one could run their classroom in a
typical flipped classroom style with the guided activities in class, and videos outside of class. You can access
the Video Playlist by clicking on this link. Similar links are located throughout the book.
Each chapter has at least one modeling activity for students to work through using the functions from
that chapter. In my experience being able to read scenarios, parse them, and write the equations that
describe the situation is the most difficult thing for students to do in Calculus. Many students also struggle
with algebra, but with advances in technology gaps in those skills are becoming easier to overcome. To
demonstrate this many of the modeling activities chosen are standard calculus problems that can be solved
graphically via a technology such as Desmos.
I have included many of the traditional approaches to solving equations but some of them come with
the title “Extension.” I typically use these as extra credit opportunities for students to work through, or as
supplemental work for advanced students. If your audience is primarily math majors, future teachers, or
students who want to excel at standardized testing these would be a valuable inclusion.

I hope you enjoy!

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Contents

1 Functions 5
1.1 What is a function, and how can I tell? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Functions and Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Function Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Average rate of change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Linear Functions 13
2.1 Introduction to Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Modeling Activity: Pond border . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Linear functions from tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4 Linear functions from graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Point slope form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Quadratic Functions 19
3.1 General form of quadratic functions and factoring . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Modeling activity: maximizing area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Vertex form of quadratic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Using the quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Extension: Proof of the quadratic formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Unit Circle Trigonometry 29

4.1 The unit circle, radians, and degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2 The functions sin(x) and cos(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3 The first quadrant and reference angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 Other trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.5 Inverse Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Exponential Functions 41
5.1 Exponent rule review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.2 The structure of exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Modeling Activity: Chessboard Fable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 Modeling Activity: Student loan growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Logarithmic Functions 47
6.1 The definition of logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Logarithm Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2.1 The rule for products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2.2 The rule for quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2.3 The rule for powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3
 6.2.4 The change of base formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2.5 Logarithm summary sheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.3 Solving logarithmic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.4 Using logarithms to solve exponential equations . . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.5 Modeling with exponential functions and logarithms . . . . . . . . . . . . . . . . . . . . . . . 59

7 Triangle Trigonometry 61
7.1 Trigonometric functions and right triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2 Finding angles with inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 64
7.3 Modeling with Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.4 Sum and Difference Identities for sin(x) and cos(x) . . . . . . . . . . . . . . . . . . . . . . . . 66
7.5 Extension: Using Trigonometry for Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Polynomial Functions 69
8.1 Introduction to Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
8.2 Modeling Activity: Box optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.3 Modeling Activity: Framing a picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.4 Extension: Factoring polynomials by division . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

9 Rational Functions 79
9.1 Introduction to Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9.2 Vertical and horizontal asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
9.3 Two classic optimization questions: boxes and cans . . . . . . . . . . . . . . . . . . . . . . . . 84

Appendices

Appendix A Formula sheet for exams 89

Appendix B Links 95

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Chapter 1

Functions

Throughout this course we are going to be exploring different types of functions. As such we are going to
begin with some explorations surrounding functions in general with an eye toward answering the following
questions:

• What is a function? How can I determine if I have one or not?

• What type of notation do we typically use for functions?
• How can we represent functions? How are they related to graphs?
• Can we combine functions? What are some ways to do that?

• If a function changes my input is there a way for me to undo that change? Is there always an undo
button?
• How can we measure how fast a function is changing?

5
1.1 What is a function, and how can I tell?
Book Reference: Section 1.3
Throughout this course we are going to explore many different types of functions. In order to study
functions we should know what a function is:

Functions

A function is a relationship in which every input is matched with exactly one output.

1. Consider the following tables. Which ones are functions? Justify your answers.

Input Output Input Output Input Output

3 2 2 5 7 5
1. 2
8
5
8
2. 4
6
7
7
3. 3
5
9
15
7 11 8 5 7 5
4 20 2 3 1 25

2. Go to Desmos.com and click on Start Graphing. Desmos is a graphing calculator (like a TI-84) that is
free for everyone. We will use it frequently in this class so it may be helpful for you to download the
Desmos App on your phone or tablet. The app looks like this:

3. In the upper left of the screen you should see a big + sign. Click on it and choose to add a table.
Enter the data from the 3 tables above. What letter does Desmos use for the input? Which letter is
used for the output? What is different about the tables that are not functions?
4. Enter the following into the left sidebar on Desmos. Determine which ones are functions and which
ones are not functions.

(a) x2 − y = x + 1
(b) y 2 = x3 + 7
(c) 7y − 2x + 5 = 0
(x−2)2 (y−3)2
(d) 2 + 3 =3

5. Can you come up with a test that will allow us to check if a given graph is a function? What would
you call this test? Write down your test in your notebook and prepare to share it with the class.
6. We are going to do a class activity around functions on Desmos. Look for the class code to join Card
Sort: Functions.

6
1.2 Function Notation
Book Reference: Section 1.4
Now that we know what a function is our next task is to understand the notation that we use to describe
functions. For example consider the description of the line below:

y = 2x + 5.

Plug this into Desmos to verify that this is a line then type in the notation below:

f (x) = 2x + 5.

Desmos recognizes these as the same because it understand the notation that we use for functions. Instead
of using the letter y for our output we use the expression f (x) as this gives us a better description. The
letter f is the name of our function. We could use anything we want here to help us describe our function,
or differentiate from other functions. The letter x is telling us what letter we are using for our variable. See
the chart below:

Name of Variable Formula

=
function for input for output

= Multiply by
f (x) 2x + 5
2 then add 5

1. Given then function f (x) = 2x + 5 we can use Desmos to find different outputs quickly. If I want to
know that output of the function when I have an input of 2, I would find f (2). Type this into Desmos
and it should tell you that f (2) = 9. Try using Desmos to find the outputs at x = 7, x = 292, and
x = e5 .

2. Describe a scenario that could be modeled by the function f (x) = 2x + 5. That is, describe some
inputs and outputs that would fit this function.
3. For each of the functions below identify the name of the function, the variable for the input, and the
formula for the output. Then describe the formula in words and give a real world scenario that could
be described by the function.
5
(a) g(x) = 2 − 3x
(b) A(x) = (x + 5)(x)
(c) T (h) = 60h

4. Use Desmos to find the out puts of the functions g, A, and T , at x = −1, 7, 37 and give an interpretation
of these outputs based on the description of your functions above.
5. We are going to do another class activity around functions on Desmos. Look for the class code to join
Function Carnival which will help us connect graphs of functions to real world scenarios.

7
1.3 Functions and Graphs
Book Reference: Section 1.6
When we are looking at the graph of a function we are seeing all of the possible input-output pairs. We
traditionally use the x-axis to represent our inputs and the y-axis to represent our outputs. For example,
the graph below is from the function f (x) = − 32 x + 2. Observe that the point (3, 0) is on the graph of this
function. This tells us that if we plug in x = 3 we will get an output of 0.

f (x) = − 32 x + 2

1. Using the graph above find f (6), the output corresponding to an input of 6.
2. Using the graph above solve the equation f (x) = 2. That is, find the input that gives you an output
of 2.

3. Use the graph above to answer the following questions

(a) Find f (1) (c) Solve f (x) = 5

(b) Find f (3) (d) Solve f (x) = −1

8
1.4 Function Composition
Book Reference: Section 5.1
When we are using function notation we often use the letter x to represent our variable. For example we
can consider the function defined by
f (x) = x2 − 5
We see that our function is named f , and whatever we have as an input we are going to square it and then
subtract 5. We usually replace x with a number but we can also plug in many other expressions into our
function. For example:

• f (2) = (2)2 + 5 • f (2 + 3) = (2 + 3)2 + 5

• f (h) = (h)2 + 5 • f (x + 3) = (x + 3)2 + 5

Any time we see an x that represents our input, so we always replace ALL of the variables with our
chosen input.

1. Let g(x) = x2 + 2x − 1. Determine the following:

(a) g(2) (c) g(x2 )

(b) g(h) (d) g(x + 3)

Now we are ready to see a new operation that we call function composition. Informally students
often think about this as plugging one function into another.

Function Composition

We use the following notation for function composition:

(g ◦ f )(x) = g(f (x))

Example: Let g(x) = x3 and f (x) = 2x + 5. Then

(g ◦ f )(x) = g(f (x)) = g(2x + 5) = (2x + 5)3

2. Let g(x) = x3 , and f (x) = 4x. Determine the following:

(a) g(2x) (c) (f ◦ g)(x)

(b) f (x + 5) (d) (g ◦ f )(x)

3. Let g(x) = x2 − 2x, and f (x) = 3 − 2x. Determine the following:

(a) (g ◦ f )(x) (c) (f ◦ g)(x)

(b) (g ◦ g)(x) (d) (f ◦ f )(x)

9
1.5 Inverse Functions
Book Reference: Section 5.2
So far we have seen that functions take an input and change it through a series of operations. What
we want to know now is: Can these changes be undone? Is there some way to undo whatever a function does?

Example:

• Pick a Function: • Undo the function:

• f(x) = 2x + 5 • Subtract 5
• Multiply by 2 • Divide by 2
x−5
• Then add 5 • g(x) = 2

The two functions f and g above are called inverse functions because they “undo” each other.

Inverse functions

We say that two functions f (x) and g(x) are inverses if:

• (f ◦ g)(x) = x and
• (g ◦ f )(x) = x.

In this case we write g(x) = f −1 (x) and say that f and g are invertible.

x−5
1. Use function composition to verify that f (x) = 2x + 5 and g(x) = 2 are inverses.
2. Consider the function f (x) = mx+b where we take an input, multiply by m, and then add b. Determine
the inverse for this function.
3. Use Desmos to graph f (x) = mx + b and its inverse that you found in 2. Type them into Desmos with
the letters m and b and add the sliders when it prompts you to add them. Move around the values
of m and b using the sliders. What interesting properties do you notice? Record your observations in
your notebook to prepare for an in class discussion.
4. As we see in our daily lives not all actions can be undone. For example, completely burning a piece of
firewood is something that cannot be undone. Similarly with functions we cannot always undo them.
The list of functions below are all examples of functions that do not have inverses. Use Desmos to
graph them and look for some patterns that may indicate why they are not invertible.

(a) f (x) = x2 (c) h(x) = x3 − 2x + 1

x+5
(b) g(x) = 5x2 (d) p(x) = −|x| + 4

Hint 1: Using our observations about the graphs of inverses, what would the inverse of these functions
look like? Would they be functions?

Hint 2: The property that we are looking for is called One to One, and we can use something called
the Horizontal line test to determine it. Prepare your observations for an in class discussion.

10
1.6 Average rate of change
Book Reference: Section 2.1
Throughout the calculus sequence we are often studying how fast quantities are changing. One of the
ways we can measure how fast a function is changing is called the average rate of change.

Average rate of change

The average rate of change of a function f on an interval [a, b] is defined to be

f (b) − f (a)
b−a

1. When I walk my dogs in the evening we typically walk 34 of a mile and the entire walk takes about
15 minutes. What is my average walking speed during this trip? Describe how the average rate of
change formula applies to this situation. What would the function f be measuring? What are a and
b measuring?
2. The two tables below give input and output values for functions f and g.

Input Output Input Output

1 2 1 3
f (x) : 2
3
5
8
g(x) : 2
3
6
11
4 11 4 18
5 14 5 27

Determine the average rate of change of f and g on the intervals

(a) [1, 2] (c) [3, 5]

(b) [2, 4] (d) [1, 5]

3. Find the average rate of change of the following functions on the interval [−2, 2].

1
(a) f (x) = 2x + 5 (c) h(x) = x+1
(b) g(x) = x2 − 1 (d) p(x) = 14

4. I am drinking a 12 ounce cup of coffee and notice that 2 minutes after getting my cofee I have 10
ounces of coffee remaining. Then after 5 minutes I have 4 ounces of coffee remaining. How fast am I
drinking my coffee, on average, between 2 and 5 minutes after receiving it? How fast am I drinking,
on average, in the first 5 minutes? Include appropriate units on your answer.

11
12
Chapter 2

Linear Functions

The first type of function we are going to study is a linear function. Our goal is to answer the following
questions:

• What makes a function linear?

• What are the main parts of a linear function and what information can we get from them?
• What does the slope of a line tell me and how do I find it?
• How many different ways can we represent a linear function?
• What information do I need to determine an equation of a line?

• How can I find the equation of a line from a graph?

• Which real world situations can be modeled by linear functions?

13
2.1 Introduction to Linear Functions
Book Reference: Section 2.1

Linear Function

A linear function is a function of the form

f (x) = mx + b.

This form is called the slope-intercept form where m is the slope of the function and b is the
y-intercept.

Use the function f (n) = 2n + 3 for the table below

n f (n)
-1
0 3
1
2
3
4 11
1. Fill out the missing outputs in the table above.
2. Plot the points from the table above onto the axes below, the pair (0, 3) is already listed. Write down
some observations you have about these points in your notebook. Do you think these points lie on the
graph of a linear function? Why?

3. Solve the equation defined by f (x) = −1. Describe this equation in words.
4. What is f (−4)? Describe this quantity in words.
5. Determine the average rate of change of the function f on the intervals [0, 4], [2, 4], and [2, 3]. What
do you notice? Prepare your observations for an in class discussion.

14
2.2 Modeling Activity: Pond border
Book Reference: Section 2.1
Materials: pen, paper, square pattern blocks, Desmos.

Goal: Model an area scenario using linear functions. Observe and note some of the properties we have
associated with linear functions.

Scenario:

Zoey is a tile specialist at a local pool company Westfield Pool and Tile. When an inground pool is
installed she comes in and creates the border surrounding the edge of the pool. She uses 1 foot long square
tiles and completely surrounds the pool, an example is shown below:

Square pools

1. The pool in the picture above is a 2 × 2 pool and requires 12 tiles to create the border. Determine how
many tiles are needed to create the border for a 3 × 3 pool, 4 × 4 pool, up to a 7 × 7 pool.
2. Using the data you gathered above do you notice any patterns? Can you quickly determine how many
tiles it would take to create the border for an 8 × 8 pool, 9 × 9 pool? Describe in words how you would
do that.
3. How many tiles are needed for a 15 × 15 pool? Describe in words how to determine the number of
tiles. How many tiles are needed for a 50 × 50 pool?
4. Write down an equation for the number of tiles needed to create the border for an n × n pool.
5. What is the size of the biggest pool Zoey can tile around using 100 tiles?

Rectangular pools

The new trend in pools is to have a lane pool in order to swim laps. The lane pool is 6 feet wide and can
be as long as you would like it to be. Use the Desmos graph Pool Border for the following questions.

6. How many tiles are needed to border a 10 foot long lane pool?

7. How many tiles are added when the length of the lane pool increases by 1 foot?
8. Write down an equation for the number of tiles needed to create the border for a lane pool that is n
feet long.
9. What is the size of the biggest lane pool Zoey can tile around using 100 tiles?

15
2.3 Linear functions from tables
Book Reference: Section 2.1
Consider the table below giving a value n and the value of the function f evaluated at n, which we denote
f (n).

n f (n) x g(x) t h(t)

1 2 2 16 1 5
2 5 4 12 3 9
3 8 6 8 6 15
4 11 8 4 8 19
7 20 11 -2 11 25

1. Describe the data that you see in the table for f (n) above. What do you notice about the average rate
of change? What specifically makes you think this is a linear function?
2. Make a graph of the values in the table for f (n).

3. Write an equation for f (n) and describe in detail how you figured that out.
4. Calculate f (52).

5. Describe the data that you see in the table for g(x) above. How does it compare to data in the table
for f (n)? What specifically makes you think this is a linear function?
6. Make a graph of the values in the table for g(x).

7. Write an equation for g(x) and describe in detail how you figured that out.
8. Solve the equation g(x) = 34.

9. Describe the data that you see in the table for h(t) above. How does it compare to data in the table
for f (n)? g(x)? What specifically makes you think this is a linear function?
10. Write an equation for h(t) and describe in detail how you figured that out.
11. Sketch a graph of the function h(t).

12. For which values of x does f (x) = h(x)?

16
2.4 Linear functions from graphs
Book Reference: Section 2.1

Recall: A linear function can be described as f (x) = mx+b where m is the slope and b is the y-intercept.

1. Write down the equation for the function f (x). Describe in detail how you figured this out.
2. Describe specifically how you found the slope of f (x), and how you found the y-intercept of f (x).
3. In your own words describe what information the slope of the graph gives you?

4. Determine the slope of the graph of g(x). Describe in detail how you figured this out.
5. Determine the y-intercept of g(x). Describe in detail how you figured this out.

6. Write down the equation of g(x).

7. On a new set of axes plot the points (2, 5), and (4, 9).

8. Draw in the linear function that connects the two points and label it h(x).
9. Find the equation of h(x). Describe in detail how you figured this out.

10. Find the equation of the line that connects the points (3, 7) and (1, 9).

17
2.5 Point slope form
Book Reference: Section 2.1

Point-Slope form

The point-slope form of a linear function is

y − y0 = m(x − x0 ).

The quantity m is the slope of the function and (x0 , y0 ) is any point on the function.

Using the point-slope form of a linear function often results in faster and more accurate computations.
When using the slope-intercept form of a linear function we have to compute the y-intercept which can often
be challenging when the data we have is not “nice.” For the following questions it is recommended that you
try to compute some of them using both the point-slope form, and the slope-intercept form to see which
method is preferable to you.

1. Find the equation of the line with the following information:

(a) Slope: 5, Point: (1, 1) (d) Slope: −3, Point: ( 43 , π)

(b) Slope: 2, Point: (−3, 5) (e) Slope: 0, Point: (0, 5)
(c) Slope: − 12 , Point: (9, 0) (f) Slope: π, Point: (−1, 2)

2. Find the equation of the line passing through the following two points:

(a) (0, 5) and (−1, 8) (d) (0, π) and (−4, 2π)

(b) (−1, −4) and (6, 12) (e) (3, 5) and (2, 5)
(c) ( 12 , 7) and ( 32 , 10) (f) (0, 6) and (0, 8)

18
Chapter 3

Quadratic Functions

The next type of functions we are going to study are called quadratic functions and fall into a larger class
of functions that we call polynomials. Polynomials will be studied in more detail later. For now we are
interested in answering the following questions:

• What are the different forms of a quadratic function?

• How do I evaluate quadratic functions?
• How can we solve quadratic equations? Specifically how do we
– Factor;
– Use the vertex of the function;
– Use the quadratic formula (what is that?).
• What information do I need to find the equation of a quadratic function from data?
• Extension: Where does the quadratic formula come from?

19
3.1 General form of quadratic functions and factoring
Book Reference: Section 2.3

Quadratic functions

A quadratic function is a function of the form

f (x) = ax2 + bx + c

where a, b, and c are real numbers with a 6= 0. This is called the general form of a quadratic
function.

For the functions below try evaluating them at the given points in two ways: Analytically with pen and
paper, then verify your answers in Desmos.

1. Evaluate analytically and graphically:

(a) f (x) = 2x2 − 5x + 7. Find f (2). (c) h(x) = 7x2 − 3x − 10. Find h(0).
(b) g(x) = x2 + x + 2. Find g 32 .


(d) p(x) = 15x2 + 12x − 9. Find p(1).

Now we want to focus on solving quadratic equations, that is finding an answer the question: When
does f (x) = c where f (x) is a quadratic function, and c is a number. There are several methods to do
this and the first one we are going to focus on is factoring. Work through the guided questions below
to make progress toward this goal.

2. Suppose that you have to numbers A and B. If you know that A · B = 0 what can you say about either
A or B?

3. What are the solutions to the following equation. How do you know?

(x − 7)(x + 3) = 0

The property that we see in the previous questions is called the zero product property. It is extraor-
dinarily useful and the main reason why we are often interested in factoring, or writing an equation as
the product of several terms. If we want to solve an equation of the form f (x) = 0 our main strategy
is often to factor and then set each part of our product equal to zero.

4. If we expand the product in the above equation we get the following

(x − 7)(x + 3) = x2 − 4x − 21

How do you think the coefficients in the above expressions are related to one another? That is, how
are −7 and 3 related to −4 and −21?

5. Expand the following products and think about how the factored form and general form of a quadratic
function are related.

(a) f (x) = (x + 3)(x + 2) (b) g(x) = (x − 1)(x + 4)

20
 Zero-product property

Given any two real numbers A and B, if

A·B =0

then either A = 0 or B = 0.

6. Observe the following

(x + a)(x + b) = x2 + (a + b)x + ab
Does this help you find a relationship between the factored form and general form? What do you think
the relationship is?
7. When we are factoring we are looking for two numbers and we want the product of those two numbers
to be our constant term, and the sum of those two numbers to be our middle coefficient. Consider the
following ! !
x2 + x − 6 = x+ x+

Can you find two numbers that multiply to −6 and add to 1?

8. Solve the following quadratic equations by factoring:

(a) x2 − 3x − 10 = 0 (c) x2 + 7x + 12 = 0
(b) x2 − 6x − 7 = 0 (d) x2 − 3x + 2 = 0

9. Try solving the following quadratic equations by factoring. What is different about them? What
do you have to do differently? After you solve the equations analytically use Desmos to solve them
graphically.

(a) 2x2 − 6x + 4 = 0 (c) x2 + x + 1 = 3

(b) 2x2 − 5x − 3 = 0 (d) 2x2 − 8x + 1 = −5

Observe that if we have a quadratic equation that is not equal to zero, like two of the examples above,
it is to our advantage to rearrange the terms so that one side is equal to zero. Then we can factor and
use the zero product property.

When we are setting an equation equal to zero we often say that we are looking for the zeros or the
roots of the equation. Asking someone to determine these is equivalent to asking them to solve the
equation f (x) = 0.

10. Find the roots of the following equations. First try to find them analytically, then find them graphically.

(a) f (x) = x2 − 4 (d) t(x) = x2 + 13x − 30

(b) g(x) = x2 − 10x + 21 (e) r(x) = 3x2 − 13x + 12
(c) h(x) = x2 + 19x + 84 (f) p(x) = 2x2 − 5x + 3

21
3.2 Modeling activity: maximizing area
Book Reference: Section 2.3
Materials: pen, paper, scissors, string, tape, Desmos.

Goal: Model an area scenario with a quadratic function. Observe the properties of the vertex of the
associated parabola and identify the information that it gives us.

Scenario:

I have two dogs named Cayley and Barley who are very interested in the neighbor dogs. When I let them
out into my backyard they always want to run next door to find them, even in the middle of the night. Due
to this I want to create a fenced in area in my backyard so that they cannot go next door. I went to the
hardware store and bought 35 yards of fencing to create a pen in my backyard next to my house. See the
diagram below as an example.

Objective: I want my dogs to be happy so I would like to create a pen that has the largest possible area
for them to run around in. Help me find the dimensions of the pen that will have the largest possible area.

1. Begin by creating a pen on your tables using the string and tape. Record the length and width of your
pen and find the area. (Remember that the area of a rectangle is length × width). Prepare to share
your answer with the class so we can collect some data.
2. If you know the length of your pen can you find the width? How do you do that? Write down an
equation that gives you the width of the pen from the length:

Width =
3. Plot the data we have received in class in Desmos. Use the length of each rectangle as the input and
the area as the output.

4. Based on your observations from the data create the pen that you think has the largest area.
5. Find a function that fits the data you have plugged into Desmos. You should use the formula for width
that you found above.
6. What are the dimensions that produce the maximum area? What is the maximum area?

Notes:

• The function that you found above should be a quadratic function, do you see it?
• The highest (or lowest) point on the quadratic function is a point that we call the vertex. This is a
point that we are often interested in as it represents an optimal point.

22
3.3 Vertex form of quadratic functions
Book Reference: Section 2.3

Vertex form of a quadratic function

An alternative form of a quadratic function is

f (x) = a(x − h)2 + k

where a, k, and h are real numbers with a 6= 0. This is called the vertex form and the vertex of
the quadratic function is the point (h, k).

1. Compare the two quadratic functions given below, what do you notice about them? What did you do
to compare them?

f (x) = x2 − 2x + 10 g(x) = (x − 1)2 + 9

2. Compare the general and vertex forms of the following functions. Do you notice any patterns?

General Form Vertex Form

x2 − 4x + 10 (x − 2)2 + 6
x2 − 6x + 10 (x − 3)2 + 1
x2 − 8x + 10 (x − 4)2 − 6
3x2 − 6x + 6 3(x − 1)2 + 3

2
2x2 − 7x + 1 2 x − 47 − 41 8

3. For each of the following convert from vertex form to general form:

(a) (x − 4)2 + 1 (b) (x + 1)2 − 11

4. For each of the following convert from general form to vertex form:

(a) x2 + 4x + 7 (c) 2x2 − 8x + 8

(b) x2 − 6x + 5 (d) 3x2 + 5

5. Based on all of the work we have done so far we should be seeing some patterns. The general form of
a quadratic function has the form f (x) = ax2 + bx + c, and the vertex form is given above. We use the
letter a in both cases because they are always the same. Use your experiences to guess the formulas
for h and k. Prepare for an in class discussion.

h= k=

23
6. Use the examples of vertex form we have on this page to explore the following questions to prepare for
a group discussion. Use other representations of the functions (tables, graphs) to aid in your thinking.

(a) Why is the vertex form of a quadratic function of interest to us?

(b) What is the vertex and why do we call it vertex form?
(c) Are there any computations that are easier to do when the function is in vertex form?

7. Find the zeros of the following quadratic functions

(a) f (x) = 2(x − 3)2 − 4

(b) g(x) = 3(x − 7)2 + 5

8. Is the vertex of a parabola enough information to find the vertex form of the quadratic function? If
not, what other information do you need?

• If you can, find two quadratic functions that have a vertex of (0, 0).
• Find the equation of a quadratic function with vertex (0, 0) that also passes through (2, 5).

9. Find the equation of the quadratic functions given in the graphs below:

(a) (c)

(b) (d)

24
3.4 Using the quadratic formula
Book Reference: Section 2.3
So far we have seen several ways to solve quadratic functions, factoring, graphing, and vertex form. Each
of these methods gives us some insight into the behavior of quadratic functions and some understanding of
the inner workings of these parabolas. In this section we will see a formula that gives us these zeros.

Quadratic formula

Given the quadratic function f (x) = ax2 + bx + c the formula

√
−b ± b2 − 4ac
x=
2a
gives the zeros of the quadratic function. This formula above is called the quadratic formula.

This section gives no insight into why this works, but that is the focus of the following extension.

Example: Use the quadratic formula to find the zeros of the function f (x) = x2 − 10x + 16.

We begin by identify that a = 1, b = −10, and c = 16. Next we plug those values into the quadratic
formula:

√
−b ± b2 − 4ac
x=
2a p
−(−10) ± (−10)2 − 4(1)(16)
=
2(1)
√
10 ± 100 − 64
=
√2
10 ± 36
=
2
10 ± 6
=
2
10 + 6 10 − 6
= or
2 2
= 8 or 2

Use the quadratic formula to find the zeros of the following functions:

1. p(x) = 2x2 − 14x + 24 5. k(x) = 2x2 − 7x + 3

2. f (x) = 6x2 − 7x + 2 6. m(x) = x2 − 2x + 2

3. g(x) = x2 + 2x − 9 7. t(x) = x2 − 10x + 25

4. h(x) = 2x2 + 3x − 4 8. r(x) = 2x2 − 8x + 9

25
3.5 Extension: Proof of the quadratic formula
Book Reference: Section 2.3
For this activity we will be exploring the famous quadratic formula! You have probably seen this before
and may even be able to recite it from memory
√
−b ± b2 − 4ac
.
2a
However, there are usually a lot of questions that remain unanswered: Where did this come from? What
are a, b, and c? What information does this give me? Our goal is to answer those questions.

1. Below is an example of completing the square for reference. We will consider 3x2 − 6x − 5.

• Factor out the leading coefficient: 3(x2 − 2x) − 5

2 2


• Take half of our interior coefficient and square it: 2 =1
2
• Add and subtract this inside the parentheses: 3(x − 2x + 1 − 1) − 5
• Regroup your subtraction outside of the parentheses: 3(x2 − 2x + 1) − 3 − 5
• Factor and simplify: 3(x − 1)2 − 8

2. Practice this procedure on the quadratics below:

• 2x2 + 4x − 7
• 2x2 + 6x − 5
• 2x2 + 3x − 7

3. We are often interested in determining where quadratics are equal to zero. Having a quadratic in
vertex form gives us a standard way to figure out where a quadratic equals zero. In part 1 we saw that

3x2 − 6x − 5 = 3(x − 1)2 − 8

Our goal: Determine the values of x for which 3(x − 1)2 − 8 = 0.

• Bring your constant term over:

3(x − 1)2 = 8
• Divide by the leading coefficient:
8
(x − 1)2 =
3
• Take the square root: r
8
x−1=±
3
• Add over the constant: r
8
x=1±
3
4. Practice this procedure on the vertex forms you found in question 2.

26
 Now we are ready see where the quadratic formula comes from. Follow the notes below with your group
and fill in the spots where appropriate. Our goal is to answer the following question: Find values of x for
which
ax2 + bx + c = 0
We will use completing the square just like we saw in the last activity.
1. Begin by factoring out the leading coefficient:
!
2
ax + bx + c = a +c

2. Take half of our interior coefficient and square it:

 2
b
2a

3. Add and subtract this inside the parentheses:

 
b
a x2 + x + − +c
a

4. Regroup your subtraction outside of the parentheses:

 2 !  2 
2 b b b
a x + x+ −a +c
a 2a 4a2

5. Factor and simplify:

!2
b2 − 4ac
a x+ −
4a

6. What we have done so far is found the vertex form for a general quadratic. Compare this to the
quadratic formula that we have seen before
√
−b ± b2 − 4ac
2a

What similarities do you notice?

27
Now that we have our vertex form, lets see if we can determine which values of x will make it equal to
zero.
Question: For which values of x does
!2
b b2 − 4ac
a x+ − =0
2a 4a

• Bring your constant term over:

!2
b
a x+ =
2a

• Divide by the leading coefficient:

!2
b b2 − 4ac
x+ =
2a

• Take the square root: r

b b2 − 4ac
x+ =±
2a 4a2

• Add over the constant:

b
x=− ±
2a

• Finally you combine the common denominators to get your solutions:

√
−b ± b2 − 4ac
x=
2a

(a) Practice using the quadratic formula on the following functions:

• f (x) = 3x2 − 7x + 2 • h(x) = 7x2 − 14x + 3

• g(x) = 5x2 − 3x + 9 • p(x) = 5x2 − 10x + 5

28
Chapter 4

Unit Circle Trigonometry

In this chapter we will study circles, specifically the unit circle, and explore how we can use the unit circle
to define trigonometric functions. We will try to answer the following questions:

1. What is the general equation of a circle, and what is the equation of the unit circle?

2. What are radians and how are they connected to degrees?

3. How do we define the functions sin(x) and cos(x)?
4. What are some important trigonometric identities?
5. How can I use the first quadrant to determine values of sin(x) and cos(x)?

6. What are the definitions of other trigonometric functions?

29
4.1 The unit circle, radians, and degrees
Book Reference: Section 10.1
For our next topic we are going to study Trigonometry. Specifically we are going to look at the relationship
between the unit circle, angles, and right triangles. To begin we will explore the equation of circles and the
unit circle.

1. Begin by opening the Desmos graph titled Circles.

(a) Move the sliders for h, k, and r. With your groups discuss how changing the values moves the
circle around. How would you define the point (h, k)? What is the meaning of r?
(b) Move around your circle until you find a circle that is centered at the origin and goes through the
point (1, 0). What is the radius of this circle?
(c) Write down the equation of the circle you just created:

Equation of unit circle:

2. Recall that the formula for the circumference for a circle is C = 2πr where r is the radius of the circle.
Write down the circumference for the circle that you just created:

Circumference of unit circle: =

3. Using the image on the back of this page mark the point (1, 0) and then mark off 12 points, including
(1, 0), that are equally spaced around the circle.
4. Using the circumference of the circle that you found above determine the distance along the circle from
(1, 0) to each of the other 11 points that you found. Assume that you start at (1, 0) and move in a
counter-clockwise direction.
The distances that you have just written down define the Radian measure of the angles that corre-
spond to the associated points on the unit circle. Many of us are used to measuring angles in degrees
so it will benefit us to talk about the relationship between degrees and radians. Explore the Desmos
activity Radians Visually.
5. Talk to your group about how many degrees are in a circle. From our work above you should see that
there are also 2π radians in a circle. Write down your guess for the relationship between these two
quantities and prepare for a class discussion about what we discovered up to this point in class.

radians = degrees

30
 From our previous discussions you should have come up with the following relationship between degrees
and radians:
π radians = 180◦

6. From this relationship we can create two conversion factors. That means a factor we multiply by to
convert from one to the other. Write the conversion factors in the spaces below:

Radians to degrees: Degrees to radians:

7. Convert the following angle measures between radians and degrees:

(a) π (e) 120◦

(b) π
3 (f) 30◦

(c) 5π
4 (g) 270◦

(d) 11π
6 (h) 300◦

What happens if we look at negative angle measures? Angles more than 2π radians? More than 360◦ ?

8. Convert the following angle measures between radians and degrees. Then determine the corresponding
points on the unit circle.

(a) −π (d) −120◦

(b) 17π
3 (e) 480◦
−15π
(c) 4 (f) −540◦

31
32
33
4.2 The functions sin(x) and cos(x)
Book Reference: Section 10.2
For our next exploration we are going to define two function based on the x and y coordinates of the
points on the unit circle.

1. Open the Desmos graph titled Unit Circle. Use the coordinates of the points to fill out the tables
below. For each of the 12 points marked on the unit circle record the radian measure of their defined
angle and then either the x or y coordinate of that point.

Radians x-coordinate Radians y-coordinate

0 0
π π
6 6

11π 11π
6 6

2. Using the tables you created above and either graph paper or Desmos sketch the graphs of the functions
defined by these tables. Label the graph created by the x-coordinates as cos(θ) and the graph created
by the y-coordinates as sin(θ).
3. Determine the following values of the sin(θ) and cos(θ).

7π


(a) sin(π) (d) cos 3

5π −5π



(b) sin 2 (e) cos 4

(c) sin(− 5π 11π

4 ) (f) cos 2

4. Based on these definitions the points on the unit circle have the form (cos(θ), sin(θ)). Earlier we found
that the equation of the unit circle was x2 + y 2 = 1. Use these two facts to write down a trigonometric
identity:

Fundamental Trigonometric Identitiy

34
4.3 The first quadrant and reference angles
Book Reference: Section 10.2
When we study trigonometry we will see that we can relate any angle to one in the first quadrant of the
unit circle. The most common angles that we see are listed in the table below. Use our Desmos graph and
what we have already seen to fill out the table below.

π π π π
θ 0 6 4 3 2

sin(θ)

cos(θ)

√ √
1. Use your calculator to determine the decimal approximations for 23 and 22 . Replace any decimals in
the table above with these fractions. Do you notice any patterns in the table above? Can you come
up with an easy to way to remember this table? Prepare your method for a group discussion.
2. Choose angles that fall into quadrant II and study how the values of sin(θ) and cos(θ) relate to the
values in the first quadrant. How can you use the table above to evaluate sin(θ) and cos(θ) for these
angles? Go through the same process for quadrants III and IV. Write down general formulas to find
reference angles for angles in each quadrant. The first two are given here:

• Quadrant I • Quadrant III

Angle: θ Angle: θ
Reference angle: θ Reference angle:

• Quadrant II
Angle: θ • Quadrant IV
Reference angle: π − θ Angle: θ
Reference angle:

5π
3. Consider the angle θ = 3 .

(a) Which quadrant is θ in?

(b) Which angle in quadrant I does θ relate to?
(c) Is sin(θ) positive or negative?
(d) Find sin(θ) and cos(θ).
4. Use the first quadrant table above to determine the exact values for the following. Write each of the
values and identify the quadrant that each angle lies in.

5π


(a) sin 6 (e) sin (π)
2π
7π



(b) cos 6
(f) cos 3
4π
3π



(c) sin 2
(g) sin 3
3π 7π



(d) cos 4 (h) cos 4

35
4.4 Other trigonometric functions
Book Reference: Section 10.3

Trigonometric functions

There are four other trigonometric functions that we often use, these functions are called tangent,
cotangent, secant, and cosecant. The functions are defined as follows:

sin(θ) 1
tan(θ) = sec(θ) =
cos(θ) cos(θ)

cos(θ) 1
cot(θ) = csc(θ) =
sin(θ) sin(θ)

1. Use the definitions above to find the exact values of the following:

5π


(a) tan 6 (e) tan (π)
2π
7π



(b) cot 6
(f) cot 3
4π
3π



(c) sec 2
(g) sec 3
3π 7π



(d) csc 4 (h) csc 4

2. Determine all of the angles where the functions tan(θ) and sec(θ) are undefined. Why are they undefined
at these angles?
3. Determine all of the angles where the functions cot(θ) and csc(θ) are undefined. Why are they undefined
at these angles?
4. For each of the four trigonometric functions above try to sketch their graphs without using Desmos.
Use your knowledge of their domains and plot several more points to sketch your graphs.
5. Verify that your graphs are correct by plotting them in Desmos.

36
4.5 Inverse Trigonometry
In chapter 1 we learned about inverse functions and used the horizontal line test to determine if a function
was invertible. Consider the functions sin(x) and tan(x) on Desmos or on the attached handout.

1. Are the functions sin(x) and tan(x) invertible? Why?

2. Since the functions we are considering are periodic, we want to restrict the domain of these functions
so that they pass the horizontal line test. Find a restricted domain for each function so that it passes
the horizontal line test.
3. Check: Is every possible output for these functions represented in your restricted domain? If not,
choose a new restricted domain so that every output is represented. Prepare for an in class discussion.
4. We know that the graphs of inverse functions are symmetric about the line y = x. Considering the
graphs of sin(x) and tan(x) on our restricted domain, sketch a graph of their inverse function on the
axes provided by reflecting the across the line y = x.
Each of the six trigonometric functions has an inverse defined similarly to the inverses of sine and
tangent but these two will be our primary focus. Formal definitions are given below:

Inverse trigonometric functions

• The function arcsin(x) is the inverse of the sine function. It has domain [−1, 1] and range
[− π2 , π2 ]. We can define it as follows
π π
y = arcsin(x) if and only if x = sin(y) where − ≤y≤
2 2

• The function arctan(x) is the inverse of the tangent function. It has domain [−∞, ∞]
and range [− π2 , π2 ]. We can define it as follows
π π
y = arctan(x) if and only if x = tan(y) where − ≤y≤
2 2

5. Use the definition of arcsin(x) and arctan(x) along with their graphs to find the exact values of the
following quantities:

(a) arcsin(0) (e) arctan (0)

(b) arcsin(1) (f) arctan (1)
√

arcsin − 12


(c) (g) arctan − 3
√  √ 
(d) arcsin 23 (h) arctan 33

Important note: Sometimes we use the notation sin−1 (x) for arcsin(x) and tan−1 (x) for arctan(x).
It is easy to confuse what the −1 means so we will typically use arcsin(x) and arctan(x), but you may
see both notations out in the wild.

37
 Example: Solving trigonometric functions

One of the uses for inverse trigonometric functions is they give us a tool to solve equations that have
trigonometric functions in them. For example, if we have the equation

2 sin(θ) = 1

We can rewrite this equation to get

1
sin(θ) =
2
and then using our knowledge of inverse trigonometry we see that
 
1 π
θ = arcsin =
2 6

But wait! We know that there are more answers than this. Since we restricted the domain on our trig
functions to define the inverse trig functions we know have to use our understanding of trigonometry
to get all of the solutions. Since sin(x) > 0 in Quadrant I and Quadrant II we will look for solutions
in both.

• Quadrant I. We have our first solution from the arcsin function as π6 . We can also have any full
rotation around the circle so we get solutions of the form π6 + 2πk for any integer k.
• Quadrant II. Since one solution is θ = π6 in the first quadrant we know that a solution in Quadrant
II will be π − θ = π − π6 = 5π6 . Similarly we get all of the full rotations around the circle. Our
complete solutions for Quadrant II have the form 5π 6 + 2πk for integers k.

π 5π
Thus the solutions to the equation 2 sin(θ) = 1 are θ = 6 + 2πk or θ = 6 + 2πk for integers k.
6. Use the example above to solve the following trigonometric equations.
√ √
(a) 2 sin(θ) = 2 (c) tan(θ) = 3
(b) 3 sin(θ) − 1 = 0 (d) 5 tan(θ) − 5 = 0

7. Challenge: Solve the following trigonometric equations

(a) 5 sin(2θ) − 3 = 0 (c) sin2 (θ) = sin(θ)

√
(b) 3 tan(5θ − 1) = 3 (d) tan2 (θ) + tan(θ) = 2

38
y = sin(x)

y = tan(x)

39
Sketch the inverse graph for sin(x)

Sketch the inverse graph for tan(x)

40
Chapter 5

Exponential Functions

In this chapter we introduce the reader to exponential functions. We start with an refresh on our knowledge
of exponent rules and then look to answer the following questions:

1. What is the basic form of an exponential function? What information do we get from the different
parts?
2. How can we find the equation of an exponential function given two points?
3. What types of scenarios can we model with exponential functions?
4. What difficulties do we run into when we try to solve exponential functions?

41
5.1 Exponent rule review
Book Reference: Section 6.1
Rules for exponents:

• When mutliplying numbers with the same base we can add the exponents.

am · an = am+n For example: 25 · 24 = 25+4 = 29

• When dividing numbers with the same base we can subtract the exponents.

am 25
= am−n For example: = 25−4 = 21
an 24

• When we have a power to a power we can multiply the exponents.

(am )n = amn For example: (24 )3 = 24·3 = 212

• A power of a product is the product of the powers.

(ab)n = an bn For example: (2 · 3)5 = 25 · 35

• A power of a quotient is the quotient of the powers.

 a n an  5
2 25
= For example: = 5
b bn 3 3

Definitions of negative and fractional powers:

• Any nonzero number to the 0th power is 1.

a0 = 1 For example: 70 = 1

• A negative exponent represents the reciprocal.

a−n = 1 1 1
an For example: 5−2 = 2
=
5 25

• The denominators in fractional exponents represent roots.

√ √
a1/n = n
a For example: 81/3 = 3
8=2

• The numerators in fractional exponents represent powers.

√ m √ √
2 √
am/n = ( n a) = n am For example: 82/3 = 3 3
8 = 82 = 4

42
 Exponent rule practice
Use the exponent rules on the previous page to simplify the following:

1. (35 )(38 )(3−4 ) (3x + 9)3

12.
9

2. −34  3  2 4
x y x
13.
y z
3. (−2)−4

6st−4
4. 27 2/3 14.
2s−2 t2

5. (5−2 )4  y −2
15.
3z 2

1 2
6. 5−2 +


5 3/2
x2 y −4

16.
z −8
12 −2


7. 5

 2
2
−7−1 17. + 3(4)−1/2
8. 5
3−2

27x−3 y 5
(95 )2 18.
9. 9x−4 y 7
98

0 (−3)2 a5 (bc)2
10. 5π 2 + 7e 19.
(−2)3 a2 b3 c4

11. (2a3 b2 )(3ab4 )3

(x + h)3 − x3
20.
h

43
5.2 The structure of exponential functions
Book Reference: Section 6.1

Exponential function

An exponential function is one that can be written in the form

f (t) = A · bt

for some real numbers A and b such that A 6= 0, b > 0 and b 6= 1.

1. Explore the exponential function on Desmos via the link provided in PLATO. What information does
A give you about the exponential function? What information does b give you about the exponential
function?
2. The two data points (2, 126) and (3, 756) are on the graph of an exponential function. Using the model
f (t) = Abt those points give us the following information:

126 = Ab2 756 = Ab3

Use this information to first find the value of b and then once you have b find the value for A. Check
your results on Desmos.
3. Use the same process to find an exponential function of the form f (t) = Abt that connects the data
points (3, 4.5) (5, 10.125).

4. The following question is from section 6.5 #39.

According to Facebook, the number of active users of Facebook has grown significantly since its initial
launch from a Harvard dorm room in February 2004. The chart below has the approximate number
U (x) of active users, in millions, x months after February 2004. For example, the first entry (10, 1)
means that there were 1 million active users in December 2004 and the last entry (77, 500) means that
there were 500 million active users in July 2010.
Month x 10 22 34 38 44 54 59 60 62 65 67 70 72 77
Active Users
in millions 1 5.5 12 20 50 100 150 175 200 250 300 350 400 500
With the help of your classmates find a model for this data. Use the Desmos graph Facebook Data
to check for accuracy.
5. Compute the average rate of change of the following exponential functions on the given intervals. Use
the graphs of each function to make sense of this information.

1 x


(a) f (x) = 2x on [0, 1] (c) h(x) = 2 on [0, 2]
x
(d) p(x) = 8 · 23 on [1, 3]


(b) g(x) = 5 · 3x on [1, 2]

44
5.3 Modeling Activity: Chessboard Fable
Legend has it that, when chess was invented in ancient times, the emperor was so enchanted that they said to
the inventor, “Name your reward.” The inventor thought for a moment and pulled a single grain of rice out of
her pocket to show the emperor. “If you please, emperor, match my one grain of rice today on the first square
of my chessboard,” said the inventor. “Then, tomorrow place two grains on the second square, four grains
on the third square the day after, eight grains on the fourth square the day after, and so on.” The emperor
gladly agreed, thinking she was a fool for asking for a few grains of rice when she could have had gold or jewels.

Little did the emperor know they were about to look like a fool. Why?

Explore: Fill in the rest of the table to see why the emperor looks like a fool.

Square Grains on this square Total grain on the board up to this point Formula for total grains

1 1 1 1 (on board) + 1 (in hand) = 2

2 2 1+2=3

64

1. Complete the table above to gather some data on the amount of rice that is being given as a reward.
Try to make sense of the amount of rice by comparing the quantity to something else.
2. Write down a function f (x) giving the total amount of rice after rice is placed on square x.
3. How much rice does the inventor have after rice is placed on the 50th square of the board?

4. When will the inventor reach 1, 000, 000, 000, 000 grains of rice? Describe the difficulties you are having
with this computation.
5. Determine how long it will take to get the following amounts of rice:

2 grains 16 grains 152 grains 1,024 grains 20,171 grains 1,048,576 grains

6. What is the difficulty in doing these computations? Can we answer them graphically? Can we answer
them analytically?

45
5.4 Modeling Activity: Student loan growth
This modeling activity will show us how exponential functions model something that has a big effect on most
of our lives: student loans.

Materials: pen, paper, Desmos.

Goal: Understand how student loan growth can be modeled by exponential functions.

Scenario: At the end of your time at Westfield State you may find yourself with $30,000 in student
loans that have an Annual Percentage Yield (APY) of 5.2%. These numbers are close to the average student
loan debt and the current interest rate for 2018. This exploration will allow us to see how these loans grow
over time.

1. The APY on a loan gives you a normalized way to compare interest rates that takes into account the
compounding of interest. It simplifies many of our computations as we can start thinking about growth
on a yearly basis, rather than a daily, or monthly basis. If you are unfamiliar with APY take a minute
to do some research and discuss with your group.
2. Assuming that you make no payments, how much money will you owe on your student loans after 1
year?
3. Create a table in your notebook like the one shown below that gives the amount you will owe on your
loans after 1, 2, 3, 4, ..., 10 years. For each year include the amount owed and the way you computed
that amount.

Year amount owed Formula

0 $30,000
1
2
3

4. Write down an exponential function L(t) that gives the amount owed on the loan after t years if no
payments are made. Use this formula to compute how much you would owe on the loan after 15, 20,
25, and 30 years.
5. How long will it take for your amount owed to double? You can use Desmos to come up with an
approximate answer. Discuss with your group how to determine an exact answer.
6. How long will it take for your amount owed to grow to $80,000? You can use Desmos to come up with
an approximate answer. Discuss with your group how to determine an exact answer.

46
Chapter 6

Logarithmic Functions

In this chapter we are going to explore a new function called a logarithm which is the inverse of the exponential
function. We are interested in answering the following questions:

1. Why do I want to use a logarithm? What types of questions does it help me answer?

2. What is the definition of a logarithm and how does it relate to the exponential function?
3. What are the various properties of logarithms?
4. How can I use logarithms to solve exponential equations?
5. How do I solve logarithmic equations?

6. What are the various scenarios that can be modeled with exponential and logarithmic equations?

47
6.1 The definition of logarithm
Book Reference: Section 6.1 and 6.2
We have explored the connection between logarithms and exponential functions and saw that logarithms
were a tool that we could use to determine certain exponents. To explore this further we need a formal
definition of a logarithm, which is given below:

Logarithmic function

The function f (x) = logb (x) is define as

logb (x) = y if and only if by = x

The result of this is that logb (x) is an exponent, it is the power of b that gives x as a result.

Example: Consider the quantity

log2 (8)
If we let log2 (8) = y we can rewrite the equation as 2y = 8. We ask ourselves: What power of 2 makes
8? Since 23 = 8 we get
log2 (8) = 3.

Practice using the definition of the logarithm by computing the following without a calculator

1. 5.
log7 715


log2 (16)

2. 6.  
log3 (27) 8
log2/3
27
3.
7.
 
1
log5 log122 (1)
5

4. 8.
log1/2 (4) 4log4 (17)

Based on our work above we should be able to identify the logarithm as the inverse of the exponential
function. Think about what it means for functions to be inverses and complete the properties below:

• For all x we have logb (bx ) = and

• For all x > 0 we have blogb (x) = .

48
6.2 Logarithm Rules
6.2.1 The rule for products
Book Reference: Section 6.2

Exploration 1: Rule for products

For this exploration we want to determine a way to rewrite the the logarithm of a product.

logb (uw) =???

In order to determine this rule we are going to let:

logb (uw) = a logb (u) = c logb (w) = d

1. Rewrite a, c, and d using the definition of logarithms.

2. When you have three equations use some algebraic techniques to combine them into one equation.
3. Use exponent rules to come up with a relationship between a, c, and d.

4. Write a conclusion for the rule for products.

Rule for Products

logb (uw) =

Practice using the rule for products:

8


5. Evaluate: log2 3 + log2 (6) 7. Write without an exponent: log3 (t3 )
15 2



6. Evaluate: log3 4 + log3 5 + log3 (18) 8. Solve for x: log3 (x) + log3 (x + 2) = 1

49
6.2.2 The rule for quotients
Book Reference: Section 6.2

Exploration 2: Rule for quotients

For this exploration we want to determine a way to rewrite the the logarithm of a quotient.
u
logb =???
w
In order to determine this rule we are going to let:
u
logb =a logb (u) = c logb (w) = d
w
1. Rewrite a, c, and d using the definition of logarithms.
2. When you have three equations use some algebraic techniques to combine them into one equation.
3. Use exponent rules to come up with a relationship between a, c, and d.

4. Write a conclusion for the rule for quotients.

Rule for Quotients

u


logb w =

Practice using the rule for quotients:

 
5. Evaluate: log2 (160) − log2 (5) 7. Expand: log4 x
yz

6. Evaluate: log3 (100) − log3 (18) − log3 (50) 8. Solve for x: log5 (x2 − 1) − log5 (x − 1) = 2

50
6.2.3 The rule for powers
Book Reference: Section 6.2

Exploration 3: Rule for powers

For this exploration we want to determine a way to rewrite the the logarithm of a power.

logb (uw ) =???

In order to determine this rule we are going to let:

logb (uw ) = a logb (u) = c

1. Rewrite a and c using the definition of logarithms.

2. When you have the two equations use some algebraic techniques to combine them into one equation.
3. Use exponent rules to come up with a relationship between a and c.
4. Write a conclusion for the rule for powers.

Rule for powers

logb (uw ) =

Practice using the rule for powers:

5. Evaluate: 7. Expand:
x3
 
log2 (423 ) log
y4 z5
6. Evaluate:
8. Expand:
 
1
log √ log2 (xy)10
1000

51
6.2.4 The change of base formula
Book Reference: Section 6.2

Exploration 4: Changing bases of logarithms

For this exploration we want to determine a relationship between logarithms with different bases. In
general we would like to know how loga (x) is related to logb (x).

In order to determine this relationship I want you to start with the product

loga (x) · logb (a)

1. Use the rule of powers that we have discovered to rewrite the product above.
2. Use the definition of logarithms to simplify the expression you have just created.
3. Write down the relationship below:

loga (x) · logb (a) =

4. Divide your equation by logb (a) to find the change of base formula.

Change of base formula

loga (x) =

Practice using the change of base formula:

5. Convert to log: log7 (22) 7. Convert to ln: log11 (4)

6. Convert to log: log13 (25) 8. Convert to ln: log6 (8)

52
6.2.5 Logarithm summary sheet
Book Reference: Sections 6.1 and 6.2
What is a logarithm?

Logarithms are the tool we use to answer questions like: For what values of x does 3x = 152? They are
very powerful as they allow us to undo exponentiation in the same way that division undoes multiplication,
or subtraction undoes addition.

Definition: We define logarithms in the following way

logb (x) = y if and only if by = x

When you see logb (x) you should be asking yourself a question: “What power do I have to raise b to in
order to get x?” So a logarithm is an exponent.

Example: From our original question above we get:

3x = 152 if and only if log3 (152) = x

and so the precise answer to our question is log3 (152). In other words “What power do I have to raise 3 to
in order to get 152?” In this case log3 (152) is the exponent you have to put on 3 to get back 152.

Conventions: There are two special logarithms that we use a lot and are given their own notation:

ln(x) = loge (x) and log(x) = log10 (x)

Through our explorations we have seen the strong connection between logarithms, exponents and ex-
ponential functions. As such the rules that we have for logarithms are very similar to those we have for
exponents. The rules we discovered are summarized below.

Properties of Logarithms
• Inverse property: For all x we have logb (bx ) = x and for all x > 0 we have blogb (x) = x.
• Rule for Products: logb (uw) = logb (u) + logb (w)
u


• Rule for Quotients: logb w = logb (u) − logb (w)

• Rule for Powers: logb (uw ) = w logb (u)

logb (x)
• Change of Base: loga (x) = logb (a)

53
Practice using the properties of logarithms
Book Reference: Section 6.2
Expand the the following using the properties of logarithms and simplify:
 
1. log2 x8


5. log2 x128
2 −4

2. log(10x2 )
6. log(1000x3 y 5 )

3 2


3. ln ex  4
216
7. log6 x3 y
q 
100x2
4. log 3
yz 5
√ 
z
8. ln xy

Use the properties of logarithms to write the following as a single logarithm:

9. 4 ln(x) + 2 ln(y) 13. log3 (x) − 2 log3 (y)

10. log2 (x) + log2 (y) − log2 (z) 14. log2 (x) + log4 (x − 1)

1
11. 3 − log(x) 15. ln(x) + 2

12. 2 ln(x) − 3 ln(y) − 4 ln(z) 16. log7 (x) + log7 (x − 3) − 2

54
6.3 Solving logarithmic equations
Book Reference: Section 6.3
Now that we have studied the properties of logarithms we are going to see how we can use these properties
to solve logarithmic functions. For example, we will solve the equation

4 − 3 log2 (x + 1) = −8

4 − 3 log2 (x + 1) = −8
−3 log2 (x + 1) = −12 Subtract 4 from both sides
log2 (x + 1) = 4 Divide both sides by -3
4
x+1=2 Use the definition of logarithm (snail!)
x = 15 Subtract 1 from both sides

Steps for solving equations with involving logarithms

1. Isolate the logarithmic term

• Look for terms that are of the form logb (x) and try to get those terms by themselves.
• Use the properties of logarithms to create a single logarithmic term.

2. Use the definition of logarithm (remember the snail!).

• This will allow us to remove the logarithm and access the x terms we are interested in.

3. Use your known algebraic techniques to find a solution.

Example:
Solve the following equation:
12 + log3 (x + 6) + log3 (x) = 15

12 + log3 (x + 6) + log3 (x) = 15

log3 (x + 6) + log3 (x) = 3 Isolate the logarithm
log3 ((x + 6)x) = 3 Combine logarithms using product rule
x2 + 6x = 27 Use the definition of logarithm
2
x + 6x − 27 = 0 Recognize a quadratic equation
(x + 9)(x − 3) = 0 Factor
x = −9 or x = 3 Identify the solutions
-9 is not in the domain so 3 is the only solution Check that solutions make sense

55
Solve the following logarithmic equations.

1. log2 (x) = 7 6. log2 (x) − log2 (7) = 2

2. log5 (x) = 2 7. ln(8) − ln(x) + 5 = 12

3. log(5x) = 3 8. ln(7x + 15) − 9 = 46

4. log5 (2x − 1) = 2 9. log2 (x) + log2 (x + 15) = 4

5. log3 (x) + log3 (2x) = 1 10. log5 (x) + log5 (x + 20) = 3

56
6.4 Using logarithms to solve exponential equations
Book Reference: Section 6.3
Now that we have our logarithm rules we are going to explore how we can use logarithms to solve
exponential functions. We have seen previously an interest in solving the following equation:

2x = 153

We could arrive at a solution directly through the definition of logarithms but lets take a different route:

2x = 153
ln(2x ) = ln(153) Take the natural log of both sides
x · ln(2) = ln(153) Rule of powers
ln(153)
x= Use algebra
ln(2)

Steps for solving an equation involving exponential functions

1. Isolate the exponential function

• Look for terms that are of the form bx and try to get those terms by themselves.
• We do this because we are looking to use logarithms to get the variable not as an exponent.

2. Take the natural log of both sides of the equation and use the rule of powers.

• This will allow us to rewrite the equation where the variable is not in the exponent.

3. Use your known algebraic techniques to find a solution.

Example:
Solve the following equation:
7 · 3x − 15 = 34

7 · 3x − 15 = 34
7 · 3x = 49 Isolate the exponential
x
3 =7 Isolate the exponential
x
ln(3 ) = ln(7) Use natural log
x ln(3) = ln(7) Rule of powers
ln(7)
x=
ln(3)

57
 Use logarithms to solve the following exponential equations.

1. 4x = 212 6. 2000 = 1000 · 3−.01t

2. 6x = 501 7. 23x = 161−x

3. 72x = 59 8. 9 · 3x = 45x

100
4. 94x+1 = 15 9. 75 = 1+3e−2t

5. 100 · (2.3)x = 517 10. 80 = 50 + 8e−4x

Challenge questions: Use logarithms to solve the following equations.

Hint: Try to relate these to quadratic equations.

ex −e−x
11. 25x = 5x + 6 12. 2 =5

58
6.5 Modeling with exponential functions and logarithms
Book Reference: Section 6.3
Here we are going to see various scenarios that we can model using exponential and logarithmic functions.
Use the skills we have built up over this chapter to set up the exponential and/or logarithmic equations that
model the the scenarios presented in each question, then answer the proposed questions.

1. Suppose you are an ecologist studying the reintroduction of salamanders in a wooded area of New
England. You are wondering how they are affected by the predators in the area and are observing their
population over time. After you gather you data you find that you can model their population growth
by the following formula:
P (t) = 50e.00145t
where t is the number of months after the study began.

(a) How many salamanders were originally reintroduced into the area?
(b) When will the population of salamanders double?

2. Suppose that you have a portfolio of $5000 in investments that have an annual yield of 6%. The growth
of your portfolio can be modeled by the function

S(t) = 5000(1.06)t

where t is measured in years. If you want to use your savings for a $25, 000 trip to Italy, how long will
you have to wait until you have enough money?

3. According to Newton’s Law of Cooling the temperature of coffee, in degrees Fahrenheit, t minutes after
it is served is given by
T (t) = 70 + 90e−.01t

(a) Find and interpret T (0).

(b) When will the coffee be 100◦ ?
(c) If you let the coffee sit for a week, approximately what temperature will it be?

4. We are going to consider the reintroduction of wolves to Yellowstone National Park. According to the
National Park Service, the wolf population in Yellowstone National Park was 52 in 1996 and 118 in
1999. Using these data, find a function of the form N (t) = N0 ekt which models the number of wolves
t years after 1996. (Use t = 0 to represent the year 1996. Also, round your value of k to four decimal
places.) According to the model, how many wolves were in Yellowstone in 2002? (The recorded number
is 272.)
5. You are growing bacteria in a dish that has a carrying capacity of 10,000 bacteria. You began by
introducing some bacteria into the dish and found that after 1000 hours there were approximately
1200 bacteria. You are using the following equation to model their population over time where t is
measured in hours after the initial 1000
10, 000
P (t) =
1+ e−0.02(t−1000)
(a) How many bacteria will there be if you wait 1000 more hours?
(b) After how many more hours will there be 6000 bacteria?

59
6. The information entropy H, in bits, of a randomly generated password consisting of L characters is
given by H = L log2 (N ), where N is the number of possible symbols for each character in the password.
In general, the higher the entropy, the stronger the password.

(a) If a 7 character case-sensitive password is comprised of letters and numbers only, find the associ-
ated information entropy.
(b) How many possible symbol options per character is required to produce a 7 character password
with an information entropy of 50 bits?

7. The pH of a solution is a measure of its acidity or alkalinity. Specifically, pH = − log[H+ ] where [H+ ]
is the hydrogen ion concentration in moles per liter. A solution with a pH less than 7 is an acid, one
with a pH greater than 7 is a base (alkaline) and a pH of 7 is regarded as neutral.

(a) The hydrogen ion concentration of pure water is [H+ ] = 10−7 . Find its pH.
(b) Find the pH of a solution with [H+ ] = 6.3 × 10−13 .
(c) The pH of gastric acid (the acid in your stomach) is about 0.7. What is the corresponding
hydrogen ion concentration?

8. Chemical systems known as buffer solutions have the ability to adjust to small changes in acidity to
maintain a range of pH values. Buffer solutions have a wide variety of applications from maintaining
a healthy fish tank to regulating the pH levels in blood.

Blood is a buffer solution. When carbon dioxide is absorbed into the bloodstream it produces carbonic
acid and lowers the pH. The body compensates by producing bicarbonate, a weak base to partially
neutralize the acid. The equation which models blood pH in this situation
 
800
pH = 6.1 + log
x

where x is the partial pressure of carbon dioxide in arterial blood, measured in torr. Find the partial
pressure of carbon dioxide in arterial blood if the pH is 7.4.

60
Chapter 7

Triangle Trigonometry

This chapter is devoted to the connection between our six trigonometric functions and right triangles. We
will build these relationships, connect them to the unit circle and put them into context. Specifically we are
guided by the following questions:

1. How can we introduce a triangle into the unit circle, and how do the trigonometric functions fit into
that triangle?
2. What happens if we move from the unit circle and instead consider a circle of radius r?
3. How can we use the trigonometric functions to determine missing information in a right triangle?

4. What types of situations can I model with trigonometric functions?

5. Do these new relationships give us access to any trigonometric identities?

61
7.1 Trigonometric functions and right triangles
Book Reference: Section 10.2
Return to our Unit Circle graph in Desmos. Click on the folder icon next to Triangle to make the green
triangle appear on the unit circle.

1. Use the radius slider on Desmos to change the radius of the circle. How can you use the coordinates
of the unit circle points to determine the lengths of the sides of the green triangle? How does the size
of the triangle change as you change the radius? For example, how much bigger does the triangle get
if the circle has radius 3, compared to radius 1?

2. Use your knowledge of sin(θ) and cos(θ) to determine the lengths of the two legs of the triangle below.

3. In relation to the angle θ we can name the sides of a right triangle as follows.

Use the two triangles on this page to represent sin(θ) and cos(θ) as the ratio of the sides of a right
triangle. Use the letters a, h, o, for adjacent, hypotenuse, and opposite.

sin(θ)
4. Using the fact that tan(θ) = cos(θ) and your answers from questions 3, represent tan(θ) as the ratio of
the sides of a right triangle.
5. Write down a familiar mnemonic device to remember these ratios and give a detailed explanation
describing how you determined them.
6. Use the definition of sec(θ), csc(θ), and cot(θ) to represent each as the ratio of the sides of a right
triangle.

62
 7. Use SohCahToa and the Pythagorean Theorem to determine the value of all six trigonometric functions
evaluated at θ.

8. Use your knowledge of trigonometry and the pythagorean theorem to determine the lengths of all of
the sides of the triangle. What are the measures of the other angles in this triangle?

9. Use your knowledge of trigonometry and the pythagorean theorem to determine the lengths of all of
the sides of the triangle. What are the measures of the other angles in this triangle?

10. Use SohCahToa and the Pythagorean Theorem to determine the value of all six trigonometric functions
evaluated at θ. What is the angle measure of θ?

63
7.2 Finding angles with inverse trigonometric functions
Book Reference: Section 10.6
We will often have the side lengths of a triangle without any angle measurements. We can use inverse
trigonometry to help us find the angles of such a triangle.

1. If you don’t remember the two inverse trigonometry functions we have studied go back to the unit
circle trigonometry chapter and do a quick review with your group.
2. In the previous section we considered the triangle below. Work through the guided questions to
determine the measure of the angle θ

(a) Which trigonometric function relates the sides and angle in the triangle above?
(b) Find the value of this trigonometric function evaluated at θ.
(c) Use an appropriate inverse trig function to solve the equation above for θ.
(d) Do we have to consider the general solution to this equation? Why? Prepare for an in class
discussion.

3. Find all missing angle and side measurements for the triangles below.

(a) (c)

(b) (d)

64
7.3 Modeling with Trigonometry
Book Reference: Chapter 11
Modeling with Trigonometry

1. A 12-foot ladder is going to be leaned against a wall. The distance between the base of the ladder and
the wall is 5 feet. How high up will the ladder make contact with the wall?
2. A wire is stretched from the top of a building to the ground. The wire meets the ground 50 feet away
from the building and forms an angle of 30◦ with the ground. Use trigonometry to find the height of
the building.
3. A ladder is going to be leaned against a wall. The ladder will be sturdy if the angle it makes with the
π
wall is between 12 and π6 .

(a) How high up can a 10 foot ladder make contact with the wall while still remaining sturdy?
(b) A 15 foot ladder has its footing set 7 feet from the wall. Is the ladder sturdy?

4. I am building shelves in my office that will be 12 inches deep. In order to support the shelves I want
to place a support that runs from the front of the shelf and meets the wall 8 inches below the back of
the shelf forming a right triangle. What angle should I cut the supports so that they sit flush against
both the shelf and the wall?

5. This question is related to activity 3.4.5 from Active Calculus:

A trough is being constructed by bending a 4 × 24 (measured in feet) rectangular piece of sheet metal.
Two symmetric folds 2 feet apart will be made parallel to the longest side of the rectangle so that the
trough has cross-sections in the shape of a trapezoid, as pictured below:

At what angle should the folds be made to produce a trough with volume 40 ft3 ? Use the interactive
Desmos graphic Metal Trough to help your understanding.
6. A bungee jumper plummets from a high bridge to the river below and then bounces back over and
over again. At time t seconds after her jump, her height H (in meters) above the river is given by
π 
H(t) = 100 + 75e−t/20 cos t .
4
Find her height at the times indicated below.

(a) 0 seconds (d) 4 seconds

(b) 1 second (e) 6 seconds
(c) 2 seconds (f) 8 seconds

65
7.4 Sum and Difference Identities for sin(x) and cos(x)
Book Reference: Section 10.4
Sum and Difference Identities

1. Use your knowledge about the angle sum of a triangle to label all of the missing angles below. Then
use your knowledge of the functions sin(x) and cos(x) to determine the length of the missing sides.

2. Now compare the lengths of the sides of the outer rectangle above. Use this comparison to write down
two formulas:

• Formula 1:

• Formula 2:

66
7.5 Extension: Using Trigonometry for Rotations
In the image below the point with coordinate (x, y) was rotated around the origin by an angle of θ degrees.
Following that the right triangle formed by the point (x, y) and the origin was also rotated around the origin
by θ degrees. Our goal is to determine the coordinates of the point labeled (?, ?), giving us the equations
needed to rotate images around the origin.

In order to find these coordinates, use your knowledge of triangle trigonometry to find the lengths of all
of the line segments in this image. The dotted and dashed lines will be the key to our solution. Use the
interactive version of Rotations on Desmos.

67
68
Chapter 8

Polynomial Functions

Polynomial functions are the generalization of linear and quadratic functions. In this section we will look to
answer some of the following questions:

• What is a polynomial function? How can I identify them, and what are some of the important parts
of a polynomial?
• How do I evaluate polynomial functions? How do I solve polynomial equations?
• How are the factors of a polynomial related to the roots?
• What types of scenarios can I model using polynomials?

• Extension: How can I use polynomial division to solve polynomial equations and how is it related to
synthetic division?

69
8.1 Introduction to Polynomials
Book Reference: Sections 3.1, 3.2
We have seen quadratic functions, that is functions of the form f (x) = ax2 + bx + c. These fall into a
larger class of functions that we call polynomials.

Polynomial function

A polynomial is a function of the form

p(x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 .

Here are some examples of polynomials:

a) f (x) = x2 + 5x − 1 c) h(x) = 3x7 − .75x199

b) g(x) = 17x6 − 5x3 + 2x d) m(x) = x3 − x2 + 7x − 12

1. Evaluate these polynomials at x = 0 and x = 1. What do you notice?

Some Definitions:

• The largest power of x that appears in a polynomial is called the degree of the polynomial.
• The coefficient on the largest power is called the leading coefficient.
• The leading coefficient together with the largest power of x is called the leading term.
• The number a0 is called the constant term, or constant coefficient.

2. Identify the leading term, the degree, the leading coefficient, and the constant term for each of the
polynomials listed above.

3. The following polynomials are given in their factored form. Expand them and then identify the leading
term, the degree, the leading coefficient, and the constant term. Can you identify these without fully
expanding the polynomial?

a) f (x) = x(x − 1)(x − 2) c) h(x) = (x − 1)(x + 1)(x − 2)(x + 2)

b) g(x) = (x − 2)(x + 3)(x + 7) d) m(x) = (x + 1)3 (x − 2)(x − 1)

Recall that a number a is a zero of the polynomial p if p(a) = 0. What are the zeros of these polyno-
mials? Make a conjecture regarding how the factors of a polynomial are related to the zeros.

70
4. Verify that the following values are zeros of the given polynomials.

(a) h(x) = x5 − x3 + x; a = 0 (c) q(x) = x6 + 5x5 + 6x4 ; a = −2

(b) p(x) = x3 − 2x2 − 5x + 6; a = 3 (d) r(x) = 2x3 − x2 − 2x + 1; a = 1

The Factor Theorem

Suppose p is a nonzero polynomial. The real number c is a zero of p if and only if (x − c) is a

factor of p(x).

5. Use what you know about factoring and quadratic functions to find the solutions of the polynomial
equations given below:

(a) x3 − x = 0 (c) (x − 15)(x + 12)(4x − 3) = 0

3 2
(b) x + 2x − 15x = 0 (d) (x + 5)(x2 − 6x − 7) = 0

6. In the following you are given the zeros of a polynomial. Create a polynomial that has the given zeros.
How many polynomials can you find with the given zeros? What additional information do you need
to ensure there will only be one polynomial with the given zeros?

1 3
(a) Zeros: 1, 2, −3 (c) Zeros: 2 , 11, 4
(b) Zeros: −5, 0, 4, −1 (d) Zeros: .25, π, e, 19, 644

7. Find an equation for the polynomial given in the graph below.

8. Write the following polynomials in their factored form. Uses Desmos to help.

(a) f (x) = x4 − 12x3 + 49x2 − 78x + 40 (c) h(x) = 5x3 − 3x2 − 5x + 3

(b) g(x) = 2x3 + 5x2 − 124x − 63 (d) p(x) = 12x4 − 107x3 + 75x2 + 114x − 80

71
8.2 Modeling Activity: Box optimization
Materials: pen, 8.5 × 11 sheet of paper, scissors, ruler, tape, Desmos.

Goal: Model a volume scenario with a polynomial function. Observe how we consider restricted domains
in real world situations.

Scenario: We are going to build a box out of a sheet of paper. The process for doing this is to cut
congruent squares from the corners of the paper and then fold up the sides of the box. See the diagram below.

Question: Find the dimensions of the box that has the largest possible volume.

1. Begin by creating a box by cutting out the squares and folding up the sides. Record the length, width,
and height of your box and find the volume. (Recall that Volume = length × width × height). Prepare
to share your answer with the class so we can collect some data.
2. Plot the data we have received in class in Desmos. Use the height of each box as the input and the
volume as the output.

3. How is the height of your box related to the size of the square you cut out? If you know the height of
your box can you determine the length and width? How do you do that? Write down equations that
give you the length and width of the box from the height:

Length =

Width =

4. Based on your observations from the data create the box that you think has the largest volume.

5. Find a function that fits the data you have plugged into Desmos. You should use the formulas for
length and width that you found above.
6. What is the minimum heigh of a box that you can physically create? What is the maximum height of
a box that you can physically create? How does this affect the model that you created in Desmos?
7. What are the dimensions that produce the maximum volume? What is the maximum volume?

72
8.3 Modeling Activity: Framing a picture
This activity was inspired from: http://fawnnguyen.com/got-beg/

Materials: pen, 8.5 × 11 sheets of paper (2 colors), scissors, ruler, Desmos.

Goal: We want to use our knowledge of polynomials to help us frame a picture.

Scenario: You are tasked with framing a picture with a gold frame. The frame should go all the way
around the picture and be even on all sides. The gold comes in full sheets that you will have to cut into the
appropriate sized pieces. See the image below.

1. Begin with a 5in × 7in picture and an 8in × 8in sheet of gold. Cut up the 8 × 8 sheet to fit evenly
around the 5 × 7 picture. What is the thickness of your picture frame? How did you find the thickness?
2. The next picture you have to frame is 4in × 6in and you have the same 8in × 8in sheet of gold. What
is the thickness of your picture frame? How did you find the thickness?

3. The final picture you have to frame is 8in × 10in and you have a full 8.5in × 11in sheet of gold. What
is the thickness of your frame? How did you find the thickness?
4. Discuss the method you used for finding a suitable frame for your pictures. How did your knowledge
of polynomials help in solving this problem? Prepare for an in class discussion.
5. Extension: Generalize this problem to any size picture and any size sheet of gold. That is, what
equations would you use if you have an n × m picture and a s × t sheet of gold?

73
8.4 Extension: Factoring polynomials by division
Book Reference: Section 3.2
The factor theorem showed us that the roots of a polynomial are related to its factors. Due to this
theorem we will be interested in factoring polynomials to find the zeros. The procedure that we will use for
this is called polynomial division. This works just like long division of real numbers. Here is an example
where we are dividing x3 + 4x2 − 5x − 14 by x − 2.

The division tells us that

x3 + 4x2 − 5x − 14 = (x − 2)(x2 + 6x + 7)

and so a = 2 is a zero of the polynomial x3 + 4x2 − 5x − 14. If the remainder of the division is not zero then
x − c is not a factor of the polynomial, and c is not a zero.

1. Use polynomial division for the following computations:

(a) (5x3 − 2x2 + 1) ÷ (x − 3) (c) (3x4 − 2x3 + x2 − x + 1) ÷ (x2 − 1)

(b) (x3 + 8) ÷ (x + 2) (d) (x3 + 2x2 + 3) ÷ (x2 − 3x + 2)

2. When dividing polynomials by a term (x − c) we can streamline polynomial division into a process
called synthetic division. Review the handout from our book and prepare for a class discussion
surrounding synthetic division.

3. Use synthetic division for the following computations:

(a) (5x3 − 2x2 + 1) ÷ (x − 3) (c) (x4 − x2 + 3) ÷ (x − 1)

(b) (x3 + 8) ÷ (x + 2) (d) (x3 + x2 − 2x + 7) ÷ (x + 4)

74
 Use synthetic division for the following computations:

4. (x3 + x2 − 6x + 4) ÷ (x − 1) 7. (x4 + 9x3 + 9x2 − 9x + 2) ÷ (x + 2)

5. (x3 + x + 2) ÷ (x + 1) 8. (x4 − 2x3 − 2x2 − 2x − 3) ÷ (x − 3)

6. (3x5 − 2x3 − 1) ÷ (x − 1) 9. (x3 + x2 + 3x + 9) ÷ (x + 2)

Use polynomial division to find all of the zeros of the following polynomials. One or more of the zeros
are given:

10. x3 − 6x2 + 11x − 6; c = 1 12. 2x4 + 11x3 − 23x2 − 11x + 21; c = 1, −1

11. x3 + 2x2 − 3x − 6; c = −2 13. x3 + 4x2 − 49x − 196; c = −7

75
3.2 The Factor Theorem and the Remainder Theorem 259

x2 + 6x + 7
x 2 x3 + 4x2 5x 14
3
x + 2x 2

6x2 5x
2
6x + 12x
7x 14
7x+14
0
Next, observe that the terms x3 , 6x2 and 7x are the exact opposite of the terms above them.
The algorithm we use ensures this is always the case, so we can omit them without losing any
information. Also note that the terms we ‘bring down’ (namely the 5x and 14) aren’t really
necessary to recopy, so we omit them, too.

x2 + 6x + 7
x 2 x3 +4x2 5x 14
2x2
6x2
12x
7x
14
0
Now, let’s move things up a bit and, for reasons which will become clear in a moment, copy the x3
into the last row.

x2 + 6x + 7
x 2 x3 +4x2 5x 14
2x2 12x 14
x3 6x2 7x 0
Note that by arranging things in this manner, each term in the last row is obtained by adding the
two terms above it. Notice also that the quotient polynomial can be obtained by dividing each of
the first three terms in the last row by x and adding the results. If you take the time to work back
through the original division problem, you will find that this is exactly the way we determined the
quotient polynomial. This means that we no longer need to write the quotient polynomial down,
nor the x in the divisor, to determine our answer.

2 x3 +4x2 5x 14
2x2 12x 14
x3 6x2 7x 0
260 Polynomial Functions

We’ve streamlined things quite a bit so far, but we can still do more. Let’s take a moment to
remind ourselves where the 2x2 , 12x and 14 came from in the second row. Each of these terms was
obtained by multiplying the terms in the quotient, x2 , 6x and 7, respectively, by the 2 in x 2,
then by 1 when we changed the subtraction to addition. Multiplying by 2 then by 1 is the
same as multiplying by 2, so we replace the 2 in the divisor by 2. Furthermore, the coefficients of
the quotient polynomial match the coefficients of the first three terms in the last row, so we now
take the plunge and write only the coefficients of the terms to get

2 1 4 5 14
2 12 14
1 6 7 0
We have constructed a synthetic division tableau for this polynomial division problem. Let’s re-
work our division problem using this tableau to see how it greatly streamlines the division process.
To divide x3 + 4x2 5x 14 by x 2, we write 2 in the place of the divisor and the coefficients of
x3 + 4x2 5x 14 in for the dividend. Then ‘bring down’ the first coefficient of the dividend.

2 1 4 5 14 2 1 4 5 14
#
1

Next, take the 2 from the divisor and multiply by the 1 that was ‘brought down’ to get 2. Write
this underneath the 4, then add to get 6.

2 1 4 5 14 2 1 4 5 14
# 2 # 2
1 1 6

Now take the 2 from the divisor times the 6 to get 12, and add it to the 5 to get 7.

2 1 4 5 14 2 1 4 5 14
# 2 12 # 2 12
1 6 1 6 7

Finally, take the 2 in the divisor times the 7 to get 14, and add it to the 14 to get 0.

2 1 4 5 14 2 1 4 5 14
# 2 12 14 # 2 12 14
1 6 7 1 6 7 0
78
Chapter 9

Rational Functions

When we divide polynomial functions we get a new type of function called a rational function. In this chapter
we are interested in answering the following questions:

• How do I find the zeros of a rational function?

• What information can I get when the denominator is equal to zero?

• What happens when the numerator and denominator share a zero in a rational function?
• What are asymptotes and how do I recognize them from in a rational function? Specifically we will
study

– vertical asymptotes;
– horizontal asymptotes;
• What are scenarios that can be modeled with rational functions?

79
9.1 Introduction to Rational Functions
Book Reference: Section 4.1
In the previous chapter we studied polynomial functions. If you add, subtract, or multiply two polyno-
mials the result is another polynomial, but when you divide two polynomials you get something new. This
new function, called a rational function is the focus here.

Rational function

A rational function is a function of the form

p(x)
r(x) =
q(x)

where p(x) and q(x) are polynomials.

Here are some examples of rational functions:

x x2 − 1
a) f (x) = c) h(x) =
x+1 2x + 5
1 x3 + 2x − 3
b) g(x) = d) m(x) =
x−1 x2 − x7
1. Evaluate the four functions above at x = 0 and x = 1. What do you notice? Did you run into any
problems when you plugged in these numbers? When did those problems occur?
2. Think about the following prompt for a few minutes and share with your group to prepare for an in
class discussion.

For a given rational function which points are we going to be interested in finding? What
do you think happens at these points?

Try graphing the four functions above in Desmos to help you answer these questions.
3. Use Desmos to find the zeros of the following functions. What do you notice?

(x + 2)(x − 3) x3 − 4x
a) r(x) = c) t(x) =
x+1 5x − 9x2
(x − 3)(2x − 7) x2 − 3x + 2
b) s(x) = d) w(x) = 2
x2 + 8 x + 4x + 1

4. Write down a relationship between the zeros of a rational function r(x) and its numerator p(x) and
denominator q(x). Which one is the most important for finding the zeros? Is there one that we can
ignore?

80
 Our explorations on the previous page should have led us to conclude that the numerator and denom-
inator are both important for determining the zeros of a rational function. The numerator gives us
possible zeros but we have to check them against the denominator. These ideas are made formal in
the following property:

Zero-quotient property

Given any two real numbers A and B, if

A
=0
B
then A = 0 and B 6= 0.

5. Solve the following rational equations by factoring the numerator and denominator and then using the
zero-quotient property.

(x + 2)(x2 − 4) x2 + 11x − 26
(a) =0 (c)
x + 12 (x − 2)(x − 3)
2x3 + 5x2 − 23x + 10 x2 − 3x + 2
(b) (d) =0
x2 − 2x + 5 x−2

In part (d) above you should note that the when you plug in the value x = 2 you get an output of the
form 00 . We say that this is undefined and we represent this graphically as a hole in the graph.
6. Create some rational equations that have holes in them. Can you create a function that has the form
0
0 at x = 2 but does not have a hole in the graph? Prepare to share your examples with the class.

7. Identify any zeros and holes in the following functions.

3x2 + 11x − 4 (2x − 1)(x + 7)(3x − 4)

(a) f (x) = (c) g(x) =
x2 − 16 (6x − 8)(x + 1)
x3 + x2 − 17x + 15 9x − x3
(b) h(x) = (d) r(x) =
x2 + 7x − 8 x2 − 5x

8. Compute the average rate of change of the following functions on the given intervals. Use your Desmos
graphs from question 3. to think about the average rate of change graphically to prepare for an in
class discussion.

(x + 2)(x − 3) x3 − 4x
a) r(x) = on [1, 3] c) t(x) = on [−4, −3]
x+1 5x − 9x2
(x − 3)(2x − 7) x2 − 3x + 2
b) s(x) = on [4, 8] d) w(x) = 2 on [0, 1]
x2 + 8 x + 4x + 1

81
9.2 Vertical and horizontal asymptotes
Book Reference: Section 4.1
In this section we are going to develop tools for describing what happens when our outputs get really big,
or when our inputs get really big. We use the word asymptote to describe this behavior. Graphically we
represent them as dashed lines in a graph like in the graph below.

The vertical dashed lines are called vertical asymptotes and the horizontal dashed lines are called
horizontal asymptotes. We will have a more formal definition later.

1. Use Desmos to graph the following rational functions and identify any vertical asymptotes or holes in
the graph. What are you looking for to decide when a function has a vertical asymptote or a hole, and
how can you tell the difference? How can you tell without looking at the graph?

1 x+1
(a) f (x) = (e) q(x) =
x+1 (x + 1)2 (x − 3)
1 −2x2
(b) g(x) = (f) r(x) =
(x + 1)(x − 2) −3x2 + 3
x+1 5x4 − x3
(c) h(x) = (g) s(x) =
(x + 1)(x − 3) x4 + 3
(x − 1)2 x3 + 1
(d) p(x) = (h) t(x) =
(x − 1)(3x + 4) x2 + 1

2. For each of the functions above determine whether or not they have a horizontal asymptote. If they do,
write down what it is. What are you looking for to decide when a function has a horizontal asymptote?
How can you tell without looking at the graph?

3. Before moving on write down your own definition of vertical asymptote and horizontal asymptote. To
do this, describe the behavior of the function at an asymptote.

82
 Some new notation

When we are thinking about the behavior of a function we often want to know what happens when
are inputs are close to a particular number. If we want to consider x values close to a number c we use
the notation x → c. To consider arbitrarily large numbers we use the notation x → ∞, or x → −∞
for arbitrarily large negative numbers.

Vertical and horizontal asymptotes

Consider the function f (x).

• The vertical line x = c is a vertical asymptote for f (x) if f (x) → ±∞ when x → c.

• The horizontal line y = c is a horizontal asymptote for f (x) if f (x) → c when x → ±∞.

4. Read the formal definitions with new notation above and discuss them with your group. Make con-
nections with our previous examples of asymptotes.
5. Create a summary sheet in your notebook that describes how to find important characteristics about
a rational function. An outline is given below:
Let
p(x)
r(x) =
q(x)

• r(x) has a zero when...

• r(x) has a hole when...
• r(x) has a vertical asymptote when...
• r(x) has a horizontal asymptote when...
further, the horizontal asymptote is zero when...
and the horizontal asymptote is when...

6. Find any zeros, holes, vertical asymptotes or horizontal asymptotes of the following functions. First
try to identify them without graphing, and then use Desmos to verify your answers.

x−9 x2 + 4x + 4
(a) f (x) = (c) h(x) =
(x + 4)2 2x2 − 2x − 12
x2 x3 − 1
(b) g(x) = (d) r(x) = 2
x3 + 2x2 − 15x x −1

83
9.3 Two classic optimization questions: boxes and cans
Book Reference: Section 4.3
Question 1 - Optimizing a box
A classic calculus optimization problem. In place of Calculus we will use Desmos to help us out.

Materials: pen, paper, scissors, ruler, tape, Desmos.

Goal: Find the optimal dimensions for a box.

Scenario: Your task is to make a box with no top out of paper. The only conditions on this box is that
it should have a square base and a volume of 500 in3 . Since we don’t want to waste paper your goal is to
minimize the amount of paper that is used. What are the dimensions of the box that minimize the amount
of paper?

1. Use the materials provided and create a box with the given specifications in your group. Record your
data to share with the class. From this data we will make a guess at the optimal dimensions of the
box.
2. In order to find the minimum amount of paper necessary we will need to create some equations. Let
the box have a width of x and height h. Determine the following as functions of x if possible:

• Volume: • Height:

V (x, h) = h(x) =

• Surface Area:

A(x) =

3. What values make sense to plug in for x? Can x be a positive number, or should it be negative? Is
there a maximum or minimum value for x? What happens to the surface area when x gets really big?
What happens when it gets really small? Prepare for an in class discussion.
4. Graph the function for surface area in Desmos on the appropriate domain and adjust the window to
an appropriate scale, so you can see the function. Use this graph to determine the optimal dimensions
of the box. What is the minimum surface area?
5. Extension: Suppose we want to make the box out of two types of paper, a heavier one for the base
of the box. The heavier paper costs $.0005 per square inch while the regular paper costs $.0002 per
square inch. What are the dimensions that minimize the cost of the box? What is the minimum cost
of the box?

84
 Question 2 - Optimizing a can
Another classic calculus optimization problem. In place of Calculus we will use Desmos to help us out.

Materials: pen, paper, scissors, ruler, tape, Desmos.

Goal: Find the minimum cost of making a soup can.

Scenario: Your task is to find the minimum cost of constructing a soup can. The can is to be made out
of two materials, the cost of the lid material is $.03 per square inch and the cost of the side material is $.02
per square inch. The only restriction on the soup can is that it must hold a volume of of 15 in3 .

1. Use the materials provided and create a soup can with the appropriate volume. Record the dimensions
of the can and its cost to share with the class. Describe how you found the cost of the soup can.
2. In order to find the minimum cost we will need to create some equations. Let the can have a radius of
r and height h. Determine the following as functions of r if possible:

• Volume: • Height:

V (r, h) = h(r) =

• Surface Area:
• Cost:
A(r) =
C(r) =

3. What values make sense to plug in for the radius r? What happens to the cost when the radius gets
really big? Really small?
4. Graph the cost function in Desmos on the appropriate domain and adjust the window to an appropriate
scale, so you can see the function. Use this graph to determine the minimum cost of the box. What
are the dimensions that achieve the minimum cost?

85
86
Appendices

87
Appendix A

Formula sheet for exams

When giving exams for precalculus I emphasize to the students that I do not want them to spend time
memorizing the formulas. Instead I want them to understand the meanings behind the formulas and how
to use them to solve problems. To accommodate this emphasis I give all of the formulas to the students on
the exams. While teaching this course I have collected the formulas in the book and surveyed students as
to which formulas they want. The following pages are the result and are provided to students.

89
Precalculus reference January 12, 2019
Westfield State University
Department of Mathematics

Linear functions

∆y y2 − y1 rise Point-slope form: y − y0 = m(x − x0 )

Slope: = =
∆x x2 − x1 run
Slope-intercept form: y = mx + b

Quadratic functions

General form: f (x) = ax2 + bx + c b

Equations of vertex: h = − k = f (h)
2a
√
−b ± b2 − 4ac
2
Vertex form: g(x) = a(x − h) + k Quadratic formula: x =
2a

Polynomials and rational functions

Polynomial: p(x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0

Degree: n; Leading coefficient: an ; Constant term: a0

p(x)
Rational function: r(x) = where p(x) and q(x) are polynomials.
q(x)

Rules for exponents

a n an


am · an = am+n b = bn
am a0 = 1 when a 6= 0
an = am−n

(am )n = amn a−n = 1

an
√
(ab)n = an bn a1/n = n
a

90
Exponential functions and logarithms

Exponential function Definition of logarithm

t
f (t) = A · b where b > 1 logb (x) = y if and only if by = x

Logarithm Rules

logb (uw) = logb (u) + logb (w) logb (uw ) = w logb (u)
logb (x)
logb u


= logb (u) − logb (w) loga (x) =
w logb (a)

Absolute value
(
x x≥0
|x| =
−x x < 0

Area and volume formulas

Area of a rectangle: A = l · w Volume of a box: V = l · w · h

Area of a circle: A = π · r2 Volume of a cone: V = 13 π · r2 · h

Area of a triangle: A = 12 b · h Volume of a cylinder: V = π · r2 · h

Area of a trapezoid: A = 12 (b1 + b2 ) · h Volume of a sphere: V = 43 π · r3

Circumference of a circle: C = 2π · r Volume of a prism: V = B · h

91
Trigonometry

Radians and degrees: 2π radians = 180◦

Common angles in the first quadrant

π π π π
θ 0 6 4 3 2

√ √
1 2 3
sin(θ) 0 2 2 2 1

√ √
3 2 1
cos(θ) 1 2 2 2 0

Formulas for reference angles: Given θ in [0, 2π] these formulas give the reference angle.

Quadrant I Quadrant II Quadrant III Quadrant IV

θ π−θ θ−π 2π − θ

Definitions of trigonometric functions

sin(θ) cos(θ) 1 1
tan(θ) = cot(θ) = sec(θ) = csc(θ) =
cos(θ) sin(θ) cos(θ) sin(θ)

opposite adjacent opposite

sin(θ) = cos(θ) = tan(θ) =
hypotenuse hypotenuse adjacent

Sum and difference formulas Fundamental trigonometric identity

sin2 (θ) + cos2 (θ) = 1

sin(A ± B) = sin(A) cos(B) ± cos(A) sin(B)
tan2 (θ) + 1 = sec2 (θ)

cos(A ± B) = cos(A) cos(B) ∓ sin(A) sin(B) 1 + cot2 (θ) = csc2 (θ)

Inverse trigonometric functions

Arcsine: Domain: [−1, 1], Range: [− π2 , π2 ], y = arcsin(x) if and only if x = sin(y)

Arctangent: Domain: [−∞, ∞], Range [− π2 , π2 ], y = arctan(x) if and only if x = tan(y)

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Graphs of trigonometric functions

Sine Cosecant
1
f (x) = sin(x) g(x) = csc(x) = sin(x)

Cosine Secant
1
f (x) = cos(x) g(x) = sec(x) = cos(x)

Tangent Cotangent
1
f (x) = tan(x) g(x) = cot(x) = tan(x)

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94
Appendix B

Links

• Video Playlist

• Chapter 1

– Card sort: Functions

– Function Carnival

• Chapter 2

– Pool Border

• Chapter 4

– Circles
– Radians Visually
– Unit Circle

• Chapter 5

– Facebook Data

• Chapter 7

– Unit Circle
– Metal Trough

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