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3.04 Transformations of functions

Worksheet
Translations
1

Consider the parabola y = x^{2} - 3.

a

Complete the table of values:

b

Sketch the graph of y = x^{2} - 3.

c

What is the y-intercept of the graph?

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d

When adding a constant to the equation y = x^{2}, describe the type of transformation that occurs on its graph.

2

Consider the graph of y = \dfrac{1}{x}:

a

What translation is required to shift the graph of y = \dfrac{1}{x} to get the graph of

y = \dfrac{1}{x} + 4.

b

Hence, sketch y = \dfrac{1}{x} + 4.

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Consider the graph of y = 2^{x}:

a

What translation is required to shift the graph of y = 2^{x} to get the graph of

y = 2^{x} - 5.

b

Hence, sketch y = 2^{x} - 5.

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4

A graph of y = x^{4} is shown. Sketch the curve after it has undergone a transformation resulting in the function

y = x^{4} - 2.

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a

How should you translate the graph of y = f \left( x \right) to get the graph of y = f \left( x \right) + 4 ?

b

How should you translate the graph of y=g(x) to get the graph of y=g(x + 6) ?

6

The functions f \left(x\right) and g \left(x\right) = f \left(x + k\right) have been graphed:

a

Determine the value of k.

b

Describe the transformation that occured.

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7

Describe the shift required to transform the graph of y = a^{x} to get the graph of y = a^{\left(x + 5\right)}.

8

For each of the following equations, determine what the new equation is when their graphs are moved as described:

a

The graph of y = x^{3} is moved to the right by 10 units.

b

The graph of y = 2^{x} is moved down by 9 units.

9

Consider the graph of the hyperbola y = \dfrac{2}{x}:

a

What would be the new equation if the graph was shifted upwards by 2 units?

b

What would be the new equation if the graph was shifted to the right by 9 units?

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10

Consider the graph of y = \sqrt{4 - x^{2}}:

a

What would be the new equation if the graph was translated downwards by 7 units?

b

What would be the new equation if the graph was translated to the left by 3 units?

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11

If the graph of y = x^{4} is moved to the right by 8 units and up by 6 units, what is its new equation?

12

Consider the graph of y = x^{3}:

a

Describe the required translations to shift the graph of y = x^{3} to get the graph of \\y = \left(x + 2\right)^{3} + 4.

b

Hence, sketch y = \left(x + 2\right)^{3} + 4.

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13

Consider the graph of y = \sqrt{4 - x^{2}}:

a

Describe the required translations to shift the graph of y = \sqrt{4 - x^{2}} to get the graph of y = \sqrt{4 - x^{2}} + 2.

b

Hence, sketch y = \sqrt{4 - x^{2}} + 2.

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14

Consider the graph of y = \sqrt{25 - x^{2}}:

a

Describe the required translations to shift the graph of y = \sqrt{25 - x^{2}} to get the graph of y = \sqrt{25 - \left(x + 4\right)^{2}} - 2.

b

Hence, sketch y = \sqrt{25 - \left(x + 4\right)^{2}} - 2.

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15

The graph of y = P \left(x\right) is shown. Sketch the graph of y = P\left(x\right) - 20.

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Reflections
16

Consider the point P \left(8, - 3 \right). Sketch the point that is symmetric to P \left(8, - 3 \right) with respect to:

a

The x-axis

b

The y-axis

c

The origin

17

State whether the following functions have symmetry. If so, state the line or point of symmetry.

a

y = x^{2} + 1

b

y = x + 5

18

Is the graph shown symmetric with respect to the x-axis, the y-axis, or the origin?

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19

Does the function y = 4 x^{3} have reflective symmetry?

20

For each of the following functions:

i

Find f(-x).

ii

State whether the functions is symmetrical along the x-axis, the y-axis or neither.

a
f(x)=2x^{3}-2x
b
f(x)=0.9x^{6}-7x^{2}+4
c
f(x)=x^{3}+x-3
d
f(x)={\dfrac{1}{6x^3}}
21

Consider the point P = \left( - 6 , 1\right). Find the point obtained by:

a

Reflecting P across the x-axis

b

Reflecting P across the y-axis

c

Rotating P by 180 \degree about the origin

22

Describe the symmetry of the graph of x^{2} + y^{2} = 6.

23

Suppose f is a function such that f \left( 3 \right) = 2. State a point that lies on the graph of f if:

a

The graph of y = f \left( x \right) is symmetric with respect to the origin.

b

The graph of y = f \left( x \right) is symmetric with respect to the y-axis.

c

The graph of y = f \left( x \right) is symmetric with respect to the line x = 6.

d

f is an even function.

e

f is an odd function.

24

The graph of y = P \left(x\right) is shown. Sketch the graph of y = P \left( - x \right).

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Dilations
25

Consider the function f \left( x \right) = \sqrt{x}. Write down the new function g \left( x \right) which results from scaling f \left( x \right) vertically by a factor of \dfrac{1}{3}, and scaling horizontally by a factor of \dfrac{1}{2}.

26

A function f \left(x\right) is transformed into a new function g \left(x\right) = f \left(\dfrac{x}{k}\right). If 0 < k < 1, determine what will be the effect on the graph of f \left(x\right).

27

Consider the function f \left(x\right) = \sqrt{x}.

a

Complete the table for f \left(x\right):

b

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

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c

Complete the table of values for the transformed function g \left(x\right) = f \left( 4 x\right), giving all values in exact form.

d

Sketch a graph of g \left( x \right).

e

Describe how g \left(x\right) relates to the graph of f \left(x\right).

f

Explain why the transformed function h \left(x\right) = f \left( - 4 x \right) can be undefined for the values of x in the table.

28

The graph of y = P \left(x\right) is shown. Sketch the graph of y = 2 P \left(x\right).

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Combinations of transformations
29

Suppose that the x-intercepts of the graph of y = f \left( x \right) are - 5 and 6.

Find the x-intercepts of the graph of:

a
y = f \left( x + 4 \right)
b
y = f \left( x - 4 \right)
c
y = 3 f \left( x \right)
d
y = f \left( - x \right)
30

Consider the function f \left( x \right) = x^{2} - 5. Using function notation, describe the transformation of f that will result in the function:

a
g \left( x \right) = x^{2} - 6
b
h \left( x \right) = 16 x^{2} - 5
c
k \left( x \right) = 3 x^{2} - 15
31

Consider the graph of y = x^{3}:

Sketch the curve after it has undergone transformations resulting in the function

y = - 4 \left(x + 4\right)^{3}.

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Consider the given graph:

a

Describe the transformations of the graph of y = x^{3} to the given graph.

b

Write down the equation of the given graph.

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33

Consider the graphs of f \left(x\right) = \dfrac{3}{x} and g \left(x\right) shown:

a

Write g \left(x\right) in terms of f(x) using the transformation shown in the graph.

b

State the equation of g \left(x\right).

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Consider the graphs of f \left(x\right) = 2^{x} and g \left(x\right) shown.

a

Write g \left(x\right) in terms of f(x) using the transformation shown in the graph.

b

State the equation of g \left(x\right).

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35

For the following functions, describe the transformations that have occured:

a

From f(x) = x^{4} to g(x) = - 7 \left(x + 4\right)^{4}

b

From f(x) = x^{4} to g(x) = - 5 x^{4} + 8

c

From f(x) = x^{2} to g(x) = - 10 \left(x + 8\right)^{2} - 9

36

The table below shows values that satisfy the function f \left(x\right) = \left|x\right|.

x-3-2-10123
y=f(x)3210123
g(x)=5f(x)
h(x)=-2f(x)
a

Complete the table of values for each transformation of the function f \left(x\right).

b

Sketch the graph of g \left(x\right).

c

Describe how to transform the graph of f \left(x\right) into the graph of g \left(x\right).

d

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

e

Describe how to transform the graph of f \left(x\right) into the graph of h \left(x\right).

37

Three functions have been graphed on the number plane. g \left(x\right) and h \left(x\right) are both transformations of f \left(x\right).

a

State the equation of f \left(x\right).

b

Write g \left(x\right) in terms of f(x) using the transformation shown in the graph.

c

State the equation of g \left(x\right).

d

Write h \left(x\right) in terms of f(x) using the transformation shown in the graph.

e

State the equation of h \left(x\right).

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38

Some points on the graph of the function y = f \left( x \right) are given in the table below:

\text{Original point}g(x)\text{Corresponding point}
(9, -12)g(x)=f(x)-8
(9, -12)g(x)=6f(x)
(6, -7)g(x)=f(x-5)

Complete the table by finding the corresponding points on the graph of y = g \left( x \right).

39

Suppose that \left( - 4 , 3\right) is a point on the graph of y = g \left( x \right). Find the corresponding point on the graph of:

a

y = g \left( x + 7 \right) - 6

b

y = - 6 g \left( x - 4 \right) + 6

c

y = g \left( 6 x + 1 \right)

40

The table below shows coordinates of points on the function y = f(x). By performing the given transformation on them, find the corresponding transformed points.

\text{Point}\text{Transformation}\text{Transformed point}
(-3, -1)y=f(x-5)-2
(0,3)y=\dfrac{1}{5}f(x-3)
(1, -2)y=f(3x)-2
(-3, 5)y=f\left(\dfrac{x}{2}\right)-5
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Outcomes

1.2.1.4

examine transformations of the graphs of 𝑓(𝑥), including dilations and reflections, and the graphs of 𝑦=𝑎f(𝑥) and 𝑦=𝑓(𝑏x), translations, and the graphs of 𝑦=𝑓(𝑥+𝑐) and 𝑦=𝑓(𝑥)+𝑑; 𝑎,𝑏,𝑐,𝑑 ∈ R

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