topic badge
iGCSE (2021 Edition)

4.07 Absolute value functions

Worksheet
Linear functions
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
2
3
4
x
2
4
6
8
10
12
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
4
6
8
x
-4
-3
-2
-1
1
2
3
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
2
3
4
5
x
2
4
6
8
10
12
14
16
18
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 = \left\vert \dfrac{x}{2} - 5 \right\vert

ii

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

iii

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

iv

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

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:

-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
x
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
y
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.

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.

a
y = \left\vert x - 4 \right\vert
b
y=\vert 2x \vert
c
y=\vert 3x-6 \vert
d
y = \left\vert 2 x + 10 \right\vert
e
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.

13

Graph 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 5 - x \right\vert

d

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

Quadratic functions
14

Consider the quadratic function h \left( x \right) = x^{2} + 2.

a

Sketch the graph of the parabola y = h \left( x \right).

b

Sketch the graph of the parabola y = \vert h \left( x \right) \vert.

c

Explain why there is no difference between the graphs of the functions y=h(x) and y=\vert h(x) \vert.

15

Consider the parabola y = \left(2 - x\right) \left(x + 4\right).

a

State the y-intercept.

b

State the x-intercepts.

c

Determine the coordinates of the vertex of the parabola.

d

Sketch the graph of the parabola.

e

Sketch the function y=\vert (2-x)(x+4) \vert

16

Consider the parabola y = \left(x - 3\right) \left(x - 1\right).

a

Find the y-intercept.

b

Find the x-intercepts.

c

State the equation of the axis of symmetry.

d

Find the coordinates of the turning point.

e

Sketch the graph of the parabola.

f

Sketch the graph of the function y=|(x-3)(x-1)|.

17

Consider the parabola y = x \left(x + 6\right).

a

Find the y-intercept.

b

Find the x-intercepts.

c

State the equation of the axis of symmetry.

d

Find the coordinates of the turning point.

e

Sketch the graph of the parabola.

f

Sketch the graph of the function y = \vert x(x+6) \vert.

18

Sketch the graph of the following:

a
y = \vert (x + 2)(x - 3) \vert
b
y = \vert (x - 3)(x + 1) \vert
19

Consider the function y = \left(x + 5\right) \left(x + 1\right).

a

Sketch the graph.

b

Sketch the graph of y = \vert \left(x + 5\right) \left(x + 1\right) \vert on the same set of axes.

20

Consider the equation y = \vert x^{2} - 6 x + 8 \vert.

a

Factorise the expression x^{2} - 6 x + 8.

b

Hence, or otherwise, find the x-intercepts of the quadratic function y = x^{2} - 6 x + 8

c

Find the coordinates of the turning point.

d

Sketch the graph of the function y = x^{2} - 6 x + 8.

e

Sketch the graph of the function y = \vert x^{2} - 6 x + 8 \vert.

21

Consider the function y = \vert x^{2} + x - 12 \vert.

a

Find the x-intercepts of the curve.

b

Find the y-intercept of the curve.

c

What is the equation of the vertical axis of symmetry for the parabola?

d

Find the coordinates of the vertex of the parabola.

e

Sketch the graph of y = x^{2} + x - 12.

f

Sketch the graph of y = \vert x^{2} + x - 12 \vert.

22

Consider the function y = \vert x^{2} + 4 x-1 \vert.

a

Find the y-intercept of the parabola.

b

Find the vertex of the parabola.

c

Sketch the graph of y = x^{2} + 4 x-1.

d

Sketch the graph of y = \vert x^{2} + 4 x-1 \vert.

23

Sketch the graphs of the following functions:

a
y = \vert x^{2} - 12 x + 32 \vert
b
y = \vert x^{2} + 6 x + 4 \vert
c
y = \vert 6 x - x^{2} \vert
d
y = \vert 2 x^{2} + 9 x + 9 \vert
Sign up to access Worksheet
Get full access to our content with a Mathspace account

Outcomes

0606C1.3

Understand the relationship between y = f(x) and y = |f(x)|, where f(x) may be linear, quadratic or trigonometric.

0606C2.3

Know the conditions for f(x) = 0 to have two real roots, two equal roots, no real roots. Know the related conditions for a given line to intersect a given curve, be a tangent to a given curve, not intersect a given curve.

0606C2.4B

Find the solution set for quadratic inequalities.

What is Mathspace

About Mathspace