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1.02 Sketching transformations of any function

Worksheet
Multiple transformations
1

Consider the cubic function y = - \dfrac{x^{3}}{4} + 2

a

Is the cubic increasing or decreasing from left to right?

b

Is the function more or less steep than function y = x^{3} ?

c

What are the coordinates of the point of inflection of the function?

d

Sketch the graph y = - \dfrac{x^{3}}{4} + 2

2

Consider the hyperbolic function y = \dfrac{3}{x} - 3

a

Which curve approaches positive and negative infinity more quickly, y = \dfrac{1}{x} or \\ y = \dfrac{3}{x} - 3?

b

What is the equation of the:

i

Vertical asymptote

ii

Horizontal asymptote

c

Sketch the graph of y = \dfrac{3}{x} - 3.

3

Consider the hyperbolic function y = \dfrac{3}{x - 1} - 2

a

What is the equation of the:

i

Vertical asymptote

ii

Horizontal asymptote

b

Sketch the graph y = \dfrac{3}{x - 1} - 2

Order of transformations
4

Determine the equation of the new curve after performing the following transformations:

a

The curve y = x^{3} is reflected across the x-axis and then translated 3 units up.

b

The curve y = x^{3} is translated 2 units down and then reflected across the x-axis.

5

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

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6

For each of the following functions:

i

State whether the cubic is increasing or decreasing from left to right.

ii

State whether the function is more or less steep than the function y = x^{3} .

iii

State the coordinates of the point of inflection.

iv

Sketch the graph of the functions.

a
y = \dfrac{1}{2} \left(x - 3\right)^{3}
b
y = 2 \left(x - 2\right)^{3} - 2
7

For each of the following functions:

i

State the range of the function.

ii

Sketch the graph the function.

iii

Determine the coordinates of the vertex.

a
y = 2 \left|x - 3\right|
b
y = 4 - \left| 2 x\right|
8

Consider the equation y=3\left|x+2\right|.

a

State the range of the function.

b

Sketch the graph the function.

c

State the coordinates of the vertex.

9

Consider the function y = 4^{x}. Find the equation of the new function that results from the following transformations:

a

The function is first reflected across the x-axis.

b

This new function is then multiplied by - 2.

10

State the pair of transformations on y = x^{3} that would result in y = - x^{3} - 3.

11

Consider the function y = 3^{ - x }-1.

a

Find the y-intercept of the curve y = 3^{ - x }-1.

b

Find the horizontal asymptote of the curve y = 3^{ - x }-1.

c

Sketch the graph of y = 3^{ - x }-1.

12

Write the equation when the graph of y = \log_{4} x is translated seven units downward, six units to the left, and then reflected in the x-axis.

13

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

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14

Consider the graph of y = \log_{3} x.

a

What transformation must be done to obtain the graph of y = \log_{3} \left(x + 2\right) - 4 from y = \log_{3} x.

b

Hence sketch the graph of \\ y = \log_{3} \left(x + 2\right) - 4.

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15

If the graph of y = \left|x\right| is stretched vertically by a factor of 4 and reflected across the x-axis, what is the equation of the new graph?

16

Consider the function f \left( x \right) = - \dfrac{5}{x}.

a

What transformation must be done to obtain the graph of f \left( x \right) from y = \dfrac{1}{x}.

b

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

c

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

d

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

e

Is f \left( x \right) increasing or decreasing over its domain?

17

A graph of y = x^{3} is shown:

a
Sketch the graph of y = - 4 \left(x + 4\right)^{3}.
b
Sketch the graph of y = 2 \left(x - 2\right)^{3} - 2.
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18

A graph of y = x^{3} is shown:

a

What transformation must be done to obtain the graph of y = \left(x - 2\right)^{3} - 3 from y = x^{3}?

b

Hence Sketch the graph of y = \left(x - 2\right)^{3} - 3.

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19

Consider the function y = 3 \times 2^{x} + 2.

a

Find the y-intercept of the curve.

b

Fill in the table of values for y = 3 \times 2^{x} + 2.

x- 3- 2- 10123
y
c

Find the horizontal asymptote of the curve.

d

Sketch the graph of y = 3 \times 2^{x} + 2.

20

A graph of y = \log_{3} x is shown. Sketch the function after it has undergone transformations resulting in the equation y = - 2 \log_{3} \left(x - 2\right).

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21

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?

22

If the graph of y = \left|x\right| is translated 5 units left and 3 units down, what is the equation of the new graph?

23

Consider the parabola y = 8 + 3 \left(x - 7\right)^{2}.

a

What is the vertex of the parabola?

b

The parabola is reflected across the y-axis. What will its new equation be?

c

What will be the vertex of the new parabola formed after the reflection?

24

Consider the functions F \left(x\right) = 3^{ - x } and G \left(x\right) = - 3^{x}. What are the two transformations that are required to turn the graph of F \left(x\right) into the graph of G \left(x\right)?

25

Consider the equation y = \left|4 - 2 x\right|.

a

State the range of the function.

b

Sketch the graph of the function.

c

Find the coordinates of the vertex.

26

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

a

What transformation must be done to obtain the graph of \\ y = \sqrt{25 - \left(x + 4\right)^{2}} - 2 from \\ y = \sqrt{25 - x^{2}}\text{?}

b

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

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27

Consider the function y = 2 - 4^{ - x }.

a

Determine the y-intercept of its graph.

b

Is this an increasing or decreasing function?

c

As x approaches infinity, what function value does y approach?

d

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

28

Write down the equation of the new circle after x^{2} + y^{2} = 49 is translated:

a

5 units upwards

b

5 units downwards

c

5 units to the right

d

5 units to the left and 6 units upwards

29

For each of the following functions:

i

Find the x-intercept.

ii

Complete the table of values for each function:

x\dfrac{1}{2}124
y
iii

State the equation of the vertical asymptote.

iv

Sketch the graph of the functions.

a
y = 3 \log_{2} x - 6
b
y = - \log_{2} x + 2
30

For the following functions:

i

Find the x-intercept.

ii

Complete the table of values each function:

x- 9- 3- 1- \dfrac{1}{3}
y
iii

State the equation of the vertical asymptote.

iv

Sketch the graph of the functions.

a
y = - 2 \log_{3} \left( - x \right) + 4
b
y = \log_{3} \left( - x \right) - 2
31

Write the equation corresponding to the graph of y = 7^{x} after having been translated 5 units upward, 3 units to the left then reflected in the x-axis.

32

Consider the function: y=-4\left|x-4\right|+3

a

State the domain of the function.

b

State the range of the function.

c

Sketch the graph of y=-4\left|x-4\right|+3.

33

Consider the function: y = - \dfrac{\left|x - 2\right|}{3} + 4.

a

State the domain of the function.

b

State the range of the function.

c

Sketch the graph of y = - \dfrac{\left|x - 2\right|}{3} + 4.

34

A graph of the function f \left( x \right) = 4 + \dfrac{4}{x - 5} is shown below:

a

Complete the following statements.

i

If x > 5, what value does the function approach as x approaches 5.

ii

If x < 5, what value does the function approach as x approaches 5.

iii

As x \to \infty, what does the function approach?

iv

As x \to - \infty, what does the function approach?

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b

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

c

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

35

A parabola of the form y = \left(x - h\right)^{2} + k is symmetrical across the line x = 2, and its vertex lies 6 units below the x-axis.

a

Find the equation of the parabola.

b

Sketch the graph of the parabola.

36

Determine if the following parabolas have no x-intercepts:

a

y = \left(x - 7\right)^{2} + 4

b

y = - \left(x - 7\right)^{2} + 4

c

y = - \left(x - 7\right)^{2} - 4

d

y = \left(x - 7\right)^{2} - 4

37

The graph of y = \log_{4} x has a vertical asymptote at x = 0. By considering the transformations that have taken place, state the equation of the vertical asymptote of the following functions:

a

y = 2 \log_{4} x - 4

b

y = 2 \log_{4} x

c

y = \log_{4} \left(x - 5\right)

d

y = - \log_{4} x

e

y = \log_{4} \left(x + 3\right) - 2

38

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

What transformation was applied to the graph of y = \left|x\right| to obtain the given graph?

d

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

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39

The exponential function y = 5^x has been transformed to become y = 5^{ 5 x + 4} - 2.

a

State the horizontal translation performed.

b

State the vertical translation performed.

c

State the dilation factor used.

40

Consider the function that has been graphed:

a

What transformation is applied to the graph of y = \left|x\right| to obtain the given graph?

b

Determine the equation of the graph for x \geq 0.

c

Determine the equation of the graph for x < 0.

d

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

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41

Consider the function f \left( x \right) = \left( - x \right)^{3} - 3. What transformations must be done to the graph of y = x^{3} to get the graph of f \left( x \right)?

42

If the graph of y = \left|x\right| is compressed vertically by a factor of \dfrac{1}{4}, reflected across the x-axis and translated 3 units left and 3 units up, what is the equation of the new graph?

43

Consider the following graph:

a

What transformations are applied to the graph of y = x^{3} to obtain the plotted graph?

b

Write down the equation of the curve.

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44

The function: y = - \left(x + 5\right)^{2} - 1 is generated after performing the following transformations on a certain graph:

  • Reflection across the x-axis
  • Translation 2 units to the left
  • Translation 5 units down

What is the equation of the original graph?

45

For the function y = - 10 \left(x + 8\right)^{2} - 9, state the transformations that have occurred if the initial function was y = x^{2}.

Transforming unknown functions
46

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

a

What transformation can you notice in the graph?

b

Determine the value of k.

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47

Use the graph of y = f \left( x \right) to sketch the graph of y = f \left( x \right) + 4.

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48

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

What transformation of f \left(x\right) can be used to describe g \left(x\right)?

c

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

d

What transformation of f \left(x\right) can be used to describe h \left(x\right)?

e

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

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A graph of y = f \left( x \right) is shown:

Use the graph of f \left( x \right) to graph the function g \left( x \right) = f \left( - x \right).

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50

Suppose that f is a function, and that \left(6, - 7 \right) is a point on the graph of y = f \left( x \right). If the function g is given by g \left( x \right) = f \left( x - 5 \right), find the corresponding point on the graph of \\ y = g \left( x \right).

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Suppose that f is a function, and that \left(9, - 12 \right) is a point on the graph of y = f \left( x \right).

If the function g is given by g \left( x \right) = 6 f \left( x \right), find the corresponding point on the graph of \\ y = g \left( x \right).

52

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

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53

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

a

Sketch the graph of g(x)= 2f(-x).

b

Sketch the graph of g(x) = 0.5f(2x).

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54

Consider the graph of y=f(x):

a

Sketch the graph of g(x) = 2f(x-3).

b

Sketch the graph of g(x) = -f(0.5x).

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55

Use the graph of y = f \left(x\right) to sketch the graph of y = \dfrac{1}{2} f \left(x + 4\right) - 2.

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56

Use the graph of y = f \left(x\right) to sketch the graph of y = - \dfrac{1}{2} f \left(x - 6\right) - 2.

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57

If the maximum value of y = f \left(x\right) is 6, what is the maximum value of y = \dfrac{f \left(x + 2\right)}{2} - 3?

58

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)

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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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