Kingsway High School

Algebraic functions Grade 10

Every aspect of our life depends on one or more other aspects.

 Our mass depends on our diet

 Our success depends on our diligence
 Our level of fitness depends on our exercise routine
 In Physics, the distance a body travels in a fixed time depends on its speed.

 In Mathematics one variable depends on the other variable. ,y depends on x , eg: 𝑦 = 2𝑥 + 1

We can choose some values of x, and tabulate them: Complete the table for 𝑦 = 2𝑥 + 1

x -3 -2 0 1 4

The input values (the x values) are called the DOMAIN or independent variable . 𝑥 ∈
{… … ; … … ; … … ; … … ; … … }
The output values, (the y values) are called the RANGE, or dependant variable 𝑦 ∈
{… … ; … … ; … … ; … … ; … … }
We can represent relationships in different ways.

1. TABLES AND RULES

x -2 -1 0 1 2

y = -x +2

2. ORDERED PAIRS

( -2 ; …….) ( -1 ; …….) ( 0 ; …….) ( 1 ; …….) ( 2 ; …….)

3. GRAPHICALLY

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 4. USING FUNCTIONAL NOTATION

𝑓(𝑥) = −𝑥 + 2

This means find the function value when This means find the value of x when the
x = 2 ; or when x = - 1 function value is 6

1. Find 𝑓(2) 2. Find 𝑓(−1) 3. Find x if 𝑓(𝑥) = 6

……………………….. …………………………… ……………………………….

………………………… ……………………………. ……………………………….

………………………… …………………………….. ……………………………….

The above can be represented by a set of ordered pairs:

…………………………. ……………………………… …………………………………

Worksheet 1: Functional notation

1. f x   3x  5 . Calculate:
1.1 f 2 1.2 f a  1.3 f  1 1.4. f x  h 

2. f x   1  x 2 . Calculate:
2.1 f  3 2.2 f 1  a  2.3 f  x  2 2.4 f x   f 2

3. If f x   2x 2 and 𝑔(𝑥) = −3𝑥 + 5 determine

3.1 𝑓(1) + 𝑔(2) 3.2. 𝑥 𝑖𝑓 𝑓(𝑥) = 19 3.3. 𝑥 𝑖𝑓 𝑔(𝑥) = 10

f x  h   f x 
4. If f x   x 2  3x determine
h

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Domain and Range
To be done with your educator: Write down the domain and range of each graph:

1. 2. 3.
(3;4) 4

-3
(-2;-1)
-4

Domain: …………………….. …………………………….. ………………………….

Range : ……………………. …………………………….. ………………………….

4. 5. 6.

(3;2) 4

(-1;-2) -2 2
-2
-4

Domain: …………………….. …………………………….. ………………………….

Range : ……………………. …………………………….. ………………………….

Worksheet 2: Domain and range

Determine the domain
and range of each graph

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Reading off information from a graph

1. Use the graph alongside to read off the following:

1.1. 𝑓(2) = ………………….

1.2. 𝑓(4) = ………………….

1.3. 𝑓(0) = ………………….

1.4. 𝑓(𝑥) < 0 ………………….

1.5. 𝑓(𝑥) ≥ 0 ………………….

1.2. For which values of x, as x increases, is

𝑓(𝑥) an increasing function?

……………………………………….

1.3. For which values of x, as x increases, does

𝑓(𝑥) decrease?

………………………………………...

2. Use the graph alongside to answer the following:

2.1. 𝑔(5) = ……………………

2.2. 𝑔(0) = ……………………

2.3. 𝑓𝑖𝑛𝑑 𝑥 𝑖𝑓 𝑔(𝑥) = 4 ………………

2.4. 𝑔(𝑥) ≤ 0 ……………………

2.5. 𝑔(𝑥) > 0 ……………………

2.2. For which values of x, as x increases, is

𝑔(𝑥) an increasing function?

…………………………………………….

2.3. For which values of x, as x increases, does

𝑔(𝑥) decrease?

……………………………………………..

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Worksheet 3: Reading off graphs
1. Consider the functions 𝑓(𝑥) = −𝑥 2 + 1 and 𝑔(𝑥) = 𝑥 + 1
Use the graph to answer the following: g
1.1 𝐹𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥 𝑖𝑠
𝑓(𝑥) = 0
1.2 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑔(−1) =
1.3 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑓(0) = 1
1.4 𝐹𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥 𝑖𝑠
𝑓(𝑥) = 𝑔(𝑥)
1.5 𝐹𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥 𝑖𝑠
𝑓(𝑥) ≥ 𝑔(𝑥) -1 1
1.6 𝐹𝑜𝑟 𝑤ℎ𝑖𝑐ℎ 𝑣𝑎𝑙𝑢𝑒𝑠 𝑜𝑓 𝑥 𝑖𝑠
𝑓(𝑥) > 𝑔(𝑥) f

2. The graphs of 𝑓(𝑥) = 𝑥 2 − 4 and 𝑔(𝑥) = −𝑥 2 + 1 are drawn.

Use the graph to answer the
f
following:
For which value(s) of x is:

2.1 𝑓(𝑥) = 0
2.2 𝑔(𝑥) < 0
2.3 𝑓(𝑥) ≥ 0
2.4 𝑓(𝑥) ≤ 𝑔(𝑥)
2.5 𝑔(𝑥) − 𝑓(𝑥) = 5
2.6 𝑓(𝑥). 𝑔(𝑥) ≥ 0
2.7 𝑓(𝑥). 𝑔(𝑥) < 0

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Worksheet 4: The linear function 𝑓(𝑥 ) = 𝒎𝒙 + 𝒄
Exercise 1.
Sketch the following on the same set of axes:
1.1. y  2x  3 (gradient intercept method)
1.2 x y 4 (dual intercept method)
1.3. y  x  2 (table method)

Exercise 2.
y 2  y1
Use the gradient formula m  to calculate the gradient of the line passing through the following
x 2  x1
points:
2.1. A 2;5 and B3;4 2.2. A1;7  and B 3;2
2.3. A 3;2 and B 3;4 2.4. A0;5 and B5;0

Exercise 3.
Re-write the following in the form y  mx  c :
3.1. 3y  2x  9 3.2. yx7
3.3. 2y  x  7  0 3.3. 4 x  5 y  10

Exercise 4.
Determine the equation of the lines with the given values of m and c.
4.1. m  2 and c  6 4.2. m  1 and c  10
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4.3. m  and c  0 4.4. m  p and c  q
2

Exercise 5.

Determine the gradient , the y- intercept and the equation for each of the following graphs:

5.1. 5.2.

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5.3 5.4

Worksheet 5: The quadratic function 𝑓(𝑥) = 𝒂𝒙𝟐 + 𝒒

ie: The Parabola
1. Sketch the following on the same set of axes:

A 1.1 𝑦 = 𝑥2 B 1.1 𝑦 = −𝑥 2
1.2 𝑦 = 2𝑥 2 1.2 𝑦 = −2𝑥 2
1 1
1.3 𝑦 = 𝑥2 1.3 𝑦 = − 𝑥2
2 2

2. Sketch the following on the same set of axes:

A 1.1 𝑦 = 𝑥2 − 1 B 1.1 𝑦 = −𝑥 2 − 4
1.2 𝑦 = 𝑥2 + 1 1.2 𝑦 = −𝑥 2 + 4

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C 1.1 𝑦 = −3𝑥 2 + 3 D 1.1 𝑦 = − (𝑥 2 − 4)
4
2
1.2 𝑦 = 3𝑥 − 3 1.2 𝑦 = 9 − 𝑥2

Worksheet 6: Finding the equation of the parabola

Platinum maths pg 148 Ex 17

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Worksheet 7: Application of the parabola
1
1. Draw an accurate sketch of f : x   x2  2 .
2

Hence, use the graph to show where you would read off the following solutions:

1.1. f 1
1.2. f  3
1.3. x if f x   6
1.4. f 0

2. Given f : x   x 2  4 and h : x  mx  n C

2.1. Find OA ; OB and OC

2.2. Determine the values of m and n
2.3. Find the length of AC correct to two decimal places.
B O A

2 The graph of f x    x 2  9 and g x   2 x  6 is given.

3 g
3.1. If S (4 ; p ) lies on the graph of f, find the value of p. C
P
3.2. Determine the coordinates of C, A and B D
3.3. Hence , determine the lengths of OC and AB. E
Q
3.4. Determine the co-ordinates of D. A B
3.5. Find the length of CE. N O

3.6. If the x- coordinate of N is - 1 ; find the length of PQ,

if PQN is parallel to the y axis.

4 Platinum maths : Ex 21 pg 154: 2 ; 3 ; 5

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Worksheet 8: The Hyperbola or Hyperbolic function 𝑓(𝑥) = 𝒂𝒙 + 𝒒

1. Sketch the graphs of the following, showing the asymptotes and intercepts with the axes.
4 2 𝑥
1.1 𝑦=𝑥 1.4 𝑦=𝑥+3 *1.7 𝑦=2
−6 −4
1.2 𝑦= 1.5 𝑦= +1
𝑥 𝑥
3 3
1.3 𝑦=𝑥 1.6 𝑦=𝑥−2

2. Determine the equation of the asymptotes of the following functions:

4 2
2.1 𝑦=𝑥 2.4 𝑦=𝑥+3
−6 −4
2.2 𝑦= 2.5 𝑦= +1
𝑥 𝑥
3 3
2.3 𝑦=𝑥 2.6 𝑦=𝑥−2

3. Write down the domain and range of the following functions

4 2
2.1 𝑦=𝑥 2.2 𝑦=𝑥+3
−4 3
2.3 𝑦= +1 2.4 𝑦=𝑥−2
𝑥

Worksheet 9: Finding the equation of the Hyperbola

Platinum maths : pg 149, Ex 18

Worksheet 10: The exponential function 𝑓 (𝑥 ) = 𝑎. 𝑏 𝑥 + 𝑞

1. Sketch the graphs of the following, showing the asymptotes and intercepts with the axes.
1 𝑥
1.1 𝑦 = 2𝑥 1.4 𝑦 = (3) 1.7 𝑦 = 2. 3𝑥 − 18
1.2 𝑦 = 2.3𝑥 1.5 𝑦 = 2𝑥 − 1 1.8 𝑦 = −2𝑥
1.3 𝑦 = 2−𝑥 1.6 𝑦 = 2𝑥 + 1 1.9 𝑦 = −2𝑥 + 4

2. Determine the equation of the asymptote of the following functions:

2.1 𝑦 = 2𝑥 2.2 𝑦 = 2. 3𝑥 − 18
2.3 𝑦 = 2𝑥 − 1 2.4 𝑦 = 2𝑥 + 1 2.5 𝑦 = −2𝑥 + 4

3. Write down the domain and range of the following functions

3.1 𝑦 = 2𝑥 2.2 𝑦 = 2. 3𝑥 − 18
2.3 𝑦 = 2𝑥 − 1 2.4 𝑦 = 2𝑥 + 1 2.5 𝑦 = −2𝑥 + 4

Worksheet 11: Finding the equation of the exponential function

Platinum maths: pg 151 Ex 19

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Summary : Determining the equation of the 4 functions
Linear Function 𝑦 = 𝑚𝑥 + 𝑐

 find “c” by inspection

 find gradient by inspection

𝑦 = 𝑚𝑥 + 𝑐 find gradient by using 𝑚 =
𝑦2 −𝑦1
3
𝑥2 −𝑥1
-2
 Sub a point to find the other
value

Quadratic Function 𝑦 = 𝑎𝑥 2 + 𝑞

 find q by inspection
 sub a point to find “a” 3
2
𝑦 = 𝑎𝑥 + 𝑞

OR (-1 ;-2) .

𝑦 = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 )  Sub both x intercepts

 sub a point to find “a”

. (-6 ; 3)

Hyperbolic Function 𝑎
𝑦= +𝑞
𝑥
 sub the horizontal
asymptote
𝑎  sub a point to find “a”
𝑦= +𝑞
𝑥 OR 2

 Sub 2 points and solve 3

simultaneously to find “a”
and “q”
Exponential function 𝑦 = 𝑏𝑥 + 𝑞

 sub the horizontal

asymptote . (2 ; 5)
𝑥 
𝑦 = 𝑎. 𝑏 + 𝑞 sub a point to find “b”

OR
 sub the horizontal -4
asymptote
 sub 2 points to find “a” and
“b”

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Determine the equation of the following:

Linear Function

3
-2
𝑦 = 𝑚𝑥 + 𝑐

Quadratic Function

3
2
𝑦 = 𝑎𝑥 + 𝑞

(-1 ;-2) .
OR

𝑦 = 𝑎(𝑥 − 𝑥1 )(𝑥 − 𝑥2 ) . (-6 ; 3)

Hyperbolic Function

𝑎 2
𝑦 = +𝑞
𝑥 3

Exponential function

. (2 ; 5)
𝑥
𝑦 =𝑎 +𝑞

-4

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Worksheet 12: Past papers - sketching

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Worksheet 13: Past papers - application
DBE 2015

DBE 2012

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DOE 2007

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