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Middle Years

4.06 Modulus function

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

Consider the function y = f\left(x\right) that has been graphed.

a

Find f\left(2\right).

b

Find f\left(-2\right).

c

Write a rule for the function in the form y = \left\vert ax \right\vert.

d

Write a rule for the function in the form y = a\left\vert x \right\vert.

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3

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?

4

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?

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5

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?

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6

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

7

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

8

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

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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 vertex of f \left( x \right)?

Modulus function graphs
9

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
10

For each of the following functions:

i

Does the graph of the function open upwards or downwards?

ii

State the coordinates of the vertex.

iii

Sketch the graph.

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
11

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.

e

State whether the following modulus functions would have 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

12

Graph the following 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

13

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

a

Graph the function.

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?

14

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
15

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

a

Graph the function.

b

What are the coordinates of the vertex?

c

What is the minimum value of the function?

16

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.

b

Find the gradient between the vertex and point A.

c

Hence state the equation of the function.

17

A modulus 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).

Application
18

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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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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