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1.01 Translations and dilations of known functions

Worksheet
Power functions
1

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

2

Consider the function y = - x^{5} - 5.

a

Sketch the general shape of the function y = - x^{5} - 5.

b

Sketch the graphs of y = - x^{5} and y = - x^{5} - 5 on the same number plane.

c

What is the y-intercept of the graph y = - x^{5} - 5?

3

Sketch the general shape of the curve y = x^{4}.

Reflections
4

Consider the graph of y = 6^{ - x }:

a

What transformation must be used to obtain the graph y = - 6^{ - x } from y = 6^{ - x }.

b

Given the graph of y = 6^{ - x }, sketch the graph of y = - 6^{ - x }.

c

Describe the rate of change of the graph of y = - 6^{ - x }.

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5

Consider the graph of y = 4^{x}:

a

What transformation must be done to obtain the graph of y = 4^{ - x } from y = 4^{x}.

b

Sketch the graph ofy = 4^{ - x }.

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6

Consider the graph of y = 5^{x}:

a

What transformation must be done to obtain the graph of y = - 5^{x} from \\ y = 5^{x}?

b

Sketch the graph of y = - 5^{x}.

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Translations
7

Consider the graph of y = f \left( x \right), describe the transformation required to get the graph of y = f \left( x \right) + 4.

8

Consider the function y = \left(x - 6\right)^{2}.

a

Complete the table of values:

b

What is the minimum value of y?

c

Hence state the coordinates of the vertex of the parabola.

d

How many units to the right has y = x^{2} been translated?

e

Translate the parabola y = x^{2} on the graph to sketch y = \left(x - 6\right)^{2}.

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9

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

a

Complete the table of values:

x-2-1012
y
b

Use the graph of y = x^{2} to sketch the graph of y = x^{2} - 3.

c

What is the y-value of the y-intercept of the graph y = x^{2} - 3?

d

What type of transformation occurs on the graph when adding a constant to the equation y = x^{2}?

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10

Consider the function y = x^{4} - 4.

a

Complete the table of values for y = x^{4}:

x-2-1012
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b

Use the graph of y = x^{4} to sketch the graph of y = x^{4} - 4.

c

What is the y-intercept of the graph \\ y = x^{4} - 4?

d

What type of transformation occurs on the graph when adding a constant to the equation y = x^{4}?

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11

Consider the function y = \left(x - 2\right)^{3}.

a

Complete the following table of values:

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b

Sketch the graph on a number plane.

c

What transformation of the graph y = x^{3} will result in the graph of y = \left(x - 2\right)^{3}?

12

A graph of y = x^{4} is shown. Sketch the curve after it has undergone transformations resulting in the function y = x^{4} - 2.

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13

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

a

How we can trasform the graph of \\ y = \dfrac{1}{x} into the graph of y = \dfrac{1}{x + 2}?

b

Hence sketch the graph of y = \dfrac{1}{x + 2} on a number plane.

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14

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

a

Write down the equation of the new function g \left(x\right) which is formed by evaluating f \left(x\right) + 4.

b

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

c

What transformation has the graph of f \left(x\right) undergone to become the graph of g \left(x\right).

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15

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

a

What transformation must be done to obtain the graph of y = \left(x - 2\right)^{2} from y = x^{2}?

b

Hence sketch the graph of y = \left(x - 2\right)^{2} on a number plane.

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16

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

a

What transformation must be done to obtain the graph of y = 3^{ - x } + 2 from y = 3^{ - x }?

b

Hence sketch the graph of y = 3^{ - x } + 2 on a number plane.

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17

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

Sketch the graph after it has undergone transformations resulting in the function y = x^{3} - 4.

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18

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

a

What type of transformation must be done to obtain the graph of y = 3^{x} - 4 from y = 3^{x}.

b

Sketch the graph of y = 3^{x} - 4 on a number plane.

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19

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

a

What transformation must be done to obtain the graph of y = \dfrac{1}{x} + 3 from \\ y = \dfrac{1}{x}.

b

Hence sketch the graph of y = \dfrac{1}{x} + 3 on a number plane.

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20

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

a

What transformation must be done to obtain the graph of y = \sqrt{4 - x^{2}} + 2 from the graph of y = \sqrt{4 - x^{2}}?

b

Hence sketch the graph of \\ y = \sqrt{4 - x^{2}} + 2 on a number plane.

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21

Consider the graph of y = x^{4}.

Sketch the curve after it has undergone a transformation resulting in the function \\ y = \left(x-1\right)^{4}.

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22

Consider the graph of y = - 5^{x}:

a

State the equation of the asymptote of y = - 5^{x}.

b

What would be the asymptote of \\ y = 2 - 5^{x}?

c

How many x-intercepts would \\ y = 2 - 5^{x} have?

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23

If the graph of y = \left|x\right| is translated 5 units down, what is the equation of the new graph?

24

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

a

Write g \left(x\right) in terms of f(x).

b

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

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25

Consider the functions f\left(x\right) = x^2-5 and g\left(x\right)=x^2-6. Write g(x) in terms of f(x).

26

Consider the function y = \dfrac{- 1}{x}.

a

What value cannot be substituted for x?

b

In which quadrants does y = \dfrac{- 1}{x} lie?

c

Consider y = \dfrac{- 1}{x - 4}. What value cannot be substituted for x?

d

In which quadrants does y = \dfrac{- 1}{x - 4} lie?

e

What transformation must be done to obtain the graph of y = \dfrac{- 1}{x - 4} from y = \dfrac{- 1}{x}?

27

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

a

Write g \left(x\right) in terms of f(x).

b

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

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28

Find the equation of the new cubic when the curve y = x^{3} + 2 is translated:

a

6 units up

b

6 units down

29

Consider the hyperbolic function y = \dfrac{- 2}{x - 1}

a

State whether the following indicates the position of the hyperbola's branches relative to its asymptotes?

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ii
b

What are the equations of the vertical and horizontal asymptotes?

c

Sketch the graph of y = \dfrac{- 2}{x - 1}.

30

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

a

What is the domain of the function?

b

What is the range of the function?

c

Hence, sketch the graph of y = \left|x\right| + 2.

31

Describe how to transform the graph of y = g \left(x\right) to become the graph of y = g \left(x + 6\right).

32

Consider the function that has been graphed:

a

Determine the equation of the graph for x \geq 2.

b

Determine the equation of the graph for x < 2.

c

Hence or otherwise, state the equation of the graph for all real x.

d

What transformation must be done to obtain this graph from y = \left|x\right|?

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33

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

a

State the new equation if the graph was moved downwards by 7 units?

b

State the new equation if the graph was moved to the left by 3 units?

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34

State the equation of the function obtained by translating the graph of y = x^{3} to the right by 10 units.

35

The graph of y = 4 x has been shifted up 2 units. What is its new equation?

36

What transformation must be done to obtain the graph of y = a^{\left(x + 5\right)} from y = a^{x}?

37

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

a

What would be the new equation if the graph was shifted upwards by 4 units?

b

What would be the new equation if the graph was shifted to the right by 7 units?

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38

Consider the function y = \log_{3} \left(x + 4\right).

a

As x increases, is the function increasing or decreasing?

b

Is the function more steep, less steep or equally as steep as y = \log_{3} x?

c

What is the equation of the vertical asymptote of y = \log_{3} \left(x + 4\right)?

d

Sketch the graph of y = \log_{3} \left(x + 4\right).

39

Consider the function y = \left|x - 2\right|.

a

State the domain of the function.

b

State the range of the function.

c

Sketch the graph of y = \left|x - 2\right|.

40

Consider the graph of y = x^{2} - 3:

a

What would be the new equation if the graph of y = x^{2} - 3 moved upwards by 5 units?

b

What would be the new equation if the graph of y = x^{2} - 3 moved to the right by 3 units?

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Dilations
41

What is the dilation factor of:

a
y = 4 x^{3}
b
y = 3 x^{5}
42

A parabola of the form y = a x^{2} goes through the point \left(2, - 8 \right).

a

Find the value of a.

b

Find the coordinates of the vertex.

c

Sketch the graph of the parabola.

43

The graph of y = x^{4} has been provided on the coordinate axes below. Sketch the graph of y = \dfrac{1}{2} x^{4}.

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44

Consider the original graph y = 3^{x}. The function values of the graph are multiplied by 2 to form a new graph.

a

For each point on the original graph, find the point on the new graph:

Point on original graphPoint on new graph
\left(-1,\dfrac{1}{3}\right)\left(-1,⬚\right)
\left(0,1\right)\left(0,⬚\right)
\left(1,3\right)\left(1,⬚\right)
\left(2,9\right)\left(2,⬚\right)
b

What is the equation of the new graph?

c

Sketch the graphs of y = 3^x and the new graph on the same set of axes.

d

For negative x-values, is 2 \times 3^{x} above or below 3^{x}?

e

For positive x-values, is 2 \times 3^{x} above or below 3^{x}?

45

The graph of the function f\left(x\right), contains the points P\left(-2, 4\right) and Q\left(4, -5\right). Find two points on the graph of the following functions:

a
g(x) = 2f(x)
b
h(x) = f(2x)
c
j(x) = f(0.5x)
d
k(x) = f(-x)
46

Sketch the graph of y = 3 \left|x\right|.

47

The function f \left(x\right) = x^{3} is transformed into the function g \left(x\right) = k x^{3}.

a

For each of the following values of k state the affect k will have on g(x):

i

k < - 1

ii

- 1 < k < 0

iii

0 < k < 1

iv

k > 1

b

Sketch the graph of g \left(x\right) = - \dfrac{2}{5} x^{3}.

48

Consider the functions y = \dfrac{4}{x} and y = \dfrac{4}{2x}.

a

Which graph lies further away from the coordinate axes?

b

For hyperbolas of the form y = \dfrac{k}{x}, as k increases, describe what happens to the hyperbola.

49

Of the two functions y = 2^{x} and y = 3 \times 2^{x}, which is increasing more rapidly for x > 0?

50

Consider the functions y=2^{x} and y = 2^{3x}.

a

Sketch both functions on the same number plane.

b

Which function is increasing more rapidly for x\gt1?

51

Describe the affect on f \left(x\right) if it is transformed into a new function g \left(x\right) = f \left(\dfrac{x}{k}\right).

52

A function f\left(x\right) is transformed into a new function g\left(x\right) = f\left(\dfrac{x}{k}\right). If k\gt1, what effect will this have on the graph of g(x)?

53

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

x - 3 - 2 - 1 0123
y = f \left(x\right)3210123
a

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

x - 3 - 2 - 1 0123
y = f \left(x\right)3210123
g \left(x\right) = 5 f \left(x\right)
h \left(x\right) = - 2 f \left(x\right)
b

Sketch the graphs of f \left(x\right) and g \left(x\right) on the same number plane.

c

Describe the transformation undergone by f \left(x\right) to become g \left(x\right).

d

Sketch the graphs of f \left(x\right) and h \left(x\right) on the same number plane.

e

Describe the transformation undergone by f \left(x\right) to become h \left(x\right).

54

For each of the following functions determine the function value when x = 2:

a
f \left( x \right) = \log_{2} x
b
g \left( x \right) = \log_{2} \left(x + 6\right)
c
h \left( x \right) = \log_{2} x + 6
d
p \left( x \right) = 6 \log_{2} x
55

The x-intercepts of the graph of y = f \left( x \right) are x=- 5 and x=6. Find the x-intercepts of the following functions:

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)

56

Consider the function f(x) = x^2-5. Find the following transformations in terms of x:

a
f\left(4x\right)
b
4f\left(x\right)
c
16f\left(x\right)
d
f\left(16x\right)
57

The graph of y = x^{3} has a point of inflection at \left(0, 0\right). By considering the transformations that have taken place, find the point of inflection of each cubic curve below:

a

y = \dfrac{2}{3} x^{3}

b

y = x^{3} + 3

c

y = - x^{3} + 4

58

The graph of f \left(x\right) = 9^{x} and another exponential function, g \left(x\right) is shown:

g(x) increasing at exactly the same rate as f \left(x\right), but has a different y-intercept. Write down the equation of function g \left(x\right).

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MA12-1

uses detailed algebraic and graphical techniques to critically construct, model and evaluate arguments in a range of familiar and unfamiliar contexts

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