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3.5 Transformation of Functions

Shifts, reflections, stretches, and compressions applied to a parent graph, the order they combine in, and testing a function for even or odd symmetry.

Algebra and Trigonometry 2e, 3.5 Transformation of Functions

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38 exercises

Shifting a Tabular Function Vertically

A function f( x ) f \left(x\right) is given in the table. Create a table for the function g(x)=f(x)−3. g \left(x\right) = f \left(x\right) - 3.

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3.5 Exercise 21

The function h(t)=−4.9 t 2 +30t h \left(t\right) = - 4.9 t^{2} + 30 t gives the height h h of a ball (in meters) thrown upward from the ground after t t seconds. Suppose the ball was instead thrown from the top of a 10-m building. Relate this new height function b(t) b \left(t\right) to h(t), h \left(t\right) , and then find a formula for b(t). b \left(t\right) .

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Shifting a Tabular Function Horizontally

A function f(x) f \left(x\right) is given in the table. Create a table for the function g(x)=f(x−3). g \left(x\right) = f \left(x - 3\right) .

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Interpreting Horizontal versus Vertical Shifts

The function G(m) G \left(m\right) gives the number of gallons of gas required to drive m m miles. Interpret G(m)+10 G \left(m\right) + 10 and G(m+10). G \left(m + 10\right) .

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3.5 Exercise 30

Write a formula for a transformation of the toolkit reciprocal function f( x )= 1 x f \left(x\right) = \frac{1}{x} that shifts the function’s graph one unit to the right and one unit up.

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Reflecting a Tabular Function Horizontally and Vertically

A function f(x) f \left(x\right) is given as the table. Create a table for the functions below.

  1. ⓐ g(x)=−f(x) g \left(x\right) = - f \left(x\right)
  2. ⓑ h(x)=f(−x) h \left(x\right) = f \left(− x\right)
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3.5 Exercise 34

A function f(x) f \left(x\right) is given as the table. Create a table for the functions below.

  1. ⓐ g(x)=−f(x) g \left(x\right) = - f \left(x\right)
  2. ⓑ h(x)=f(−x) h \left(x\right) = f \left(− x\right)
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Determining whether a Function Is Even, Odd, or Neither

Is the function f(x)= x 3 +2x f \left(x\right) = x^{3} + 2 x even, odd, or neither?

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3.5 Exercise 38

Is the function f(s)= s 4 +3 s 2 +7 f \left(s\right) = s^{4} + 3 s^{2} + 7 even, odd, or neither?

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Finding a Vertical Compression of a Tabular Function

A function f f is given as the table. Create a table for the function g(x)= 1 2 f(x). g \left(x\right) = \frac{1}{2} f \left(x\right) .

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3.5 Exercise 41

A function f f is given as the table. Create a table for the function g(x)= 3 4 f(x). g \left(x\right) = \frac{3}{4} f \left(x\right) .

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3.5 Exercise 43

Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.

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3.5 Exercise 47

Write a formula for the toolkit square root function horizontally stretched by a factor of 3.

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Finding a Triple Transformation of a Tabular Function

Given the table for the function f(x), f \left(x\right) , create a table of values for the function g(x)=2f(3x)+1. g \left(x\right) = 2 f \left(3 x\right) + 1.

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3.5 Exercise 50

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal shift from a vertical shift?

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3.5 Exercise 52

When examining the formula of a function that is the result of multiple transformations, how can you tell a horizontal compression from a vertical compression?

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3.5 Exercise 54

How can you determine whether a function is odd or even from the formula of the function?

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3.5 Exercise 56

Write a formula for the function obtained when the graph is shifted as described.

f(x)=| x | f \left(x\right) = \left|x\right| is shifted down 3 units and to the right 1 unit.

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3.5 Exercise 58

Write a formula for the function obtained when the graph is shifted as described.

f(x)= 1 x 2 f \left(x\right) = \frac{1}{x^{2}} is shifted up 2 units and to the left 4 units.

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3.5 Exercise 60

Describe how the graph of the function is a transformation of the graph of the original function f. f .

y=f(x+43) y = f \left(x + 43\right)

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3.5 Exercise 62

Describe how the graph of the function is a transformation of the graph of the original function f. f .

y=f(x−4) y = f \left(x - 4\right)

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3.5 Exercise 64

Describe how the graph of the function is a transformation of the graph of the original function f. f .

y=f(x)+8 y = f \left(x\right) + 8

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3.5 Exercise 66

Describe how the graph of the function is a transformation of the graph of the original function f. f .

y=f(x)−7 y = f \left(x\right) - 7

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3.5 Exercise 68

Describe how the graph of the function is a transformation of the graph of the original function f. f .

y=f(x+4)−1 y = f \left(x + 4\right) - 1

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3.5 Exercise 70

Determine the interval(s) on which the function is increasing and decreasing.

g(x)=5 (x+3) 2 −2 g \left(x\right) = 5 \left(x + 3\right)^{2} - 2

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3.5 Exercise 72

Determine the interval(s) on which the function is increasing and decreasing.

k(x)=−3 x −1 k \left(x\right) = - 3 \sqrt{x} - 1

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3.5 Exercise 80

Tabular representations for the functions f,g, f , g , and h h are given below. Write g(x) g \left(x\right) and h(x) h \left(x\right) as transformations of f(x). f \left(x\right) .

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3.5 Exercise 96

Determine whether the function is odd, even, or neither.

f(x)=3 x 4 f \left(x\right) = 3 x^{4}

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3.5 Exercise 98

Determine whether the function is odd, even, or neither.

h(x)= 1 x +3x h \left(x\right) = \frac{1}{x} + 3 x

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3.5 Exercise 100

Determine whether the function is odd, even, or neither.

g(x)=2 x 4 g \left(x\right) = 2 x^{4}

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3.5 Exercise 102

Describe how the graph of each function is a transformation of the graph of the original function f. f .

g(x)=−f(x) g \left(x\right) = - f \left(x\right)

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3.5 Exercise 104

Describe how the graph of each function is a transformation of the graph of the original function f. f .

g(x)=4f(x) g \left(x\right) = 4 f \left(x\right)

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3.5 Exercise 106

Describe how the graph of each function is a transformation of the graph of the original function f. f .

g(x)=f(5x) g \left(x\right) = f \left(5 x\right)

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3.5 Exercise 108

Describe how the graph of each function is a transformation of the graph of the original function f. f .

g(x)=f( 1 3 x ) g \left(x\right) = f \left(\frac{1}{3} x\right)

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3.5 Exercise 110

Describe how the graph of each function is a transformation of the graph of the original function f. f .

g(x)=3f( −x ) g \left(x\right) = 3 f \left(- x\right)

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3.5 Exercise 112

Write a formula for the function g g that results when the graph of a given toolkit function is transformed as described.

The graph of f(x)=|x| f \left(x\right) = \left|x\right| is reflected over the y y -axis and horizontally compressed by a factor of 1 4 \frac{1}{4} .

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3.5 Exercise 114

Write a formula for the function g g that results when the graph of a given toolkit function is transformed as described.

The graph of f(x)= 1 x 2 f \left(x\right) = \frac{1}{x^{2}} is vertically compressed by a factor of 1 3 , \frac{1}{3} , then shifted to the left 2 units and down 3 units.

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3.5 Exercise 116

Write a formula for the function g g that results when the graph of a given toolkit function is transformed as described.

The graph of f(x)= x 2 f \left(x\right) = x^{2} is vertically compressed by a factor of 1 2 , \frac{1}{2} , then shifted to the right 5 units and up 1 unit.

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.