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iGCSE (2021 Edition)

17.09 Absolute value function (Extended)

Worksheet
Key features
1

Consider the function y = \left\vert x \right\vert.

a

What values of x can be substituted into the given function?

b

What values of y could the function have?

2

What is the domain of the function f \left( x \right) = \left\vert 6 - x \right\vert?

3

Consider the function that has been graphed.

a

What is the domain of the function?

b

What is the range of the function?

-4
-3
-2
-1
1
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3
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x
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14
y
4

Consider the function y = \left\vert x - 3 \right\vert.

a

What is the lowest possible value that this function can have?

b

What is the highest possible value that this function can have?

c

What is the range of the function?

d

What is the domain of the function?

5

Consider the graph of the function f \left( x \right).

a

State the coordinates of the vertex.

b

State the equation of the line of symmetry.

c

Find the gradient of the function for

x \gt 0.

d

Find the gradient of the function for

x \lt 0.

e

Is the graph of f\left(x\right) as steep as the graph of y = \left\vert x \right\vert?

-8
-6
-4
-2
2
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x
-4
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-1
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y
6

Consider the graph of the function f \left( x \right).

a

State the coordinate of the vertex.

b

State the equation of the line of symmetry.

c

Find the gradient of the function for

x \gt 2.

d

Find the gradient of the function for

x \lt 2.

e

Is the graph of f \left( x \right) more or less steep than the graph of y = \left\vert x \right\vert?

-1
1
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x
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f(x)
7

Consider the function y = \left\vert x \right\vert - 5.

a

Does the graph of the function open upwards or downwards?

b

State the coordinate of the vertex.

c

State the equation of the line of symmetry.

d

State whether the following functions have narrower graphs than y = \left\vert x \right\vert - 5.

i

y = \dfrac{\left\vert x \right\vert}{2} - 5

ii

y = \left\vert x - 5 \right\vert

iii

y = \left\vert x \right\vert - 5

iv

y = - 4 \left\vert x \right\vert - 5

8

Is the function f \left( x \right) = \left\vert 2 x - 2 \right\vert one-to-one?

9

Consider the following graphs of y = -3x-6 and y = 3x+6:

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x
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y
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a

Rewrite the function f \left( x \right) = \left\vert 3 x + 6 \right\vert as a piecewise function of the form:

f\left(x\right) = \begin{cases} ⬚ & \text{when}\ x\lt ⬚ \\ ⬚ & \text{when}\ x\geq ⬚ \\ \end{cases}
b

What is the domain and range of f \left( x \right)? Give your answers in interval notation.

Absolute value function graphs
10

Consider the function y = \left\vert x+1 \right\vert.

a

Complete the given table.

b

Hence graph the function.

c

State the equation of the axis of symmetry.

d

State the coordinates of the vertex.

x-2-1012
y
e

Complete the given table by writing the equation and gradient for the two lines that make up the graph of the function.

EquationGradient
x \lt -1
x \gt -1
11

For each of the following absolute value functions:

i

Does the graph of the function open upwards or downwards?

ii

State the coordinates of the vertex.

iii

Sketch the graph. You may use technology.

a

y = - \left\vert x \right\vert

b

y = - \left\vert \dfrac{x}{7} \right\vert

c
y = \left\vert x - 4 \right\vert
d
y=\vert 2x \vert
e
y=-\vert x+5 \vert
f
y=\vert 3x-6 \vert
g
y = \left\vert 2 x + 10 \right\vert
h
y = \left\vert x + 2 \right\vert
12

Consider the function y = \left\vert 6 x \right\vert.

a

Does the graph of the function open upwards or downwards?

b

State the equation of the line of symmetry.

c

State the coordinates of the vertex.

d

Graph the function. You may use technology.

e

State whether the absolute value function has a narrower graph than y = \left\vert 6 x \right\vert.

i

y = - \left\vert 6 x \right\vert

ii

y = \left\vert 4 x \right\vert

iii

y = \left\vert 7 x \right\vert

iv

y = \left\vert 9 x \right\vert + 3

13

Use technology to sketch the following absolute value functions:

a

y = \left\vert 5 x \right\vert

b

y = \left\vert x - 5 \right\vert

c

y = - \left\vert 3 x - 6 \right\vert

d

y = \left\vert 5 - x \right\vert

e

y = \left\vert \dfrac{x}{3} - 2 \right\vert

f

y = 8 - \left\vert 4 x \right\vert

14

Consider the function f \left( x \right) = 11 - \left\vert x \right\vert.

a

Graph the function using technology.

b

What is the maximum value of f \left( x \right)?

c

On which interval is the function increasing?

d

On which interval is the function decreasing?

15

Consider the function y = 3 \left\vert x \right\vert - 3.

a

Complete the table.

b

Hence graph the function.

c

State the equation of the axis of symmetry.

d

What are the coordinates of the vertex?

e

What are the x-intercepts?

f

What is the y-intercept?

g

What is the gradient of the line to the left of the vertex?

h

What is the gradient of the line to the right of the vertex?

x-2-1012
y
16

Consider the function y = \left\vert x + 2 \right\vert + 5.

a

Graph the function using technology.

b

What are the coordinates of the vertex?

c

What is the minimum value of the function?

17

An absolute value function has its vertex at \left( - 1 , 3\right), and goes through the point A \left(2, 0\right).

a

Graph the function using technology.

b

Find the gradient between the vertex and point A.

c

Hence state the equation of the function.

18

Find the equation of the following absolute value functions in the form y=\vert ax+b\vert:

a
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x
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b
-1
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x
-4
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y
c
-4
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x
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y
d
-2
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x
-1
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y
Transformations of the absolute value function
19

The graph of the function y = \left\vert x \right\vert is translated horizontally to the right by 3 units. What is the equation of the resulting graph?

20

Consider the function y = \left\vert x \right\vert - 4 that has been graphed. What transformation is applied to the graph of y = \left\vert x \right\vert to obtain the graph of

y = \left\vert x \right\vert - 4?

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21

Use the graph of y = \left\vert x \right\vert to graph y = \left\vert x - 4 \right\vert + 4.

22

An absolute value function f \left( x \right) has its vertex at \left( - 4 , 0\right), and goes through the point A \left(1, - 5 \right).

a

Find the gradient between the vertex and point A.

b

Hence state the equation of the function f \left( x \right).

c

State the domain of f \left( x \right) in interval notation.

d

State the range of f \left( x \right) in interval notation.

Application
23

Michael is training for the land speed record and takes his new car for a test drive by driving straight down a closed highway and back. His distance y in kilometres from the end of the highway x minutes after he takes off is given by the function y = \left\vert 5 x - 40 \right\vert which is graphed below.

a

How far does he drive in total?

b

How long does Michael take to reach the end of the highway?

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y
c

The next day Michael goes for another drive down the same route and his distance from the end of highway is given by

y = \left\vert 4 x - 40 \right\vert which is graphed.

Is Michael driving faster or slower than the previous day?

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x
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y
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Outcomes

0607C3.6

Use of a graphic display calculator to: sketch the graph of a function, produce a table of values, find zeros, local maxima or minima, find the intersection of the graphs of functions.

0607E3.2F

Recognition of absolute value, f(x) = | ax + b |, function types from the shape of their graphs.

0607E3.3

Determination of the value of at most two of a, b, c or d in simple linear, quadratic, cubic, reciprocal, exponential, absolute value and trignometric functions.

0607E3.6

Use of a graphic display calculator to: sketch the graph of a function produce a table of values, find zeros, local maxima or minima, find the intersection of the graphs of functions.

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