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

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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?

A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

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When examining the formula of a function that is the result of multiple transformations, how can you tell a reflection with respect to the x -axis from a reflection with respect to the y -axis?

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

For a function f , substitute ( x ) for ( x ) in f ( x ) . Simplify. If the resulting function is the same as the original function, f ( x ) = f ( x ) , then the function is even. If the resulting function is the opposite of the original function, f ( x ) = f ( x ) , then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

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Algebraic

Write a formula for the function obtained when the graph of f ( x ) = x is shifted up 1 unit and to the left 2 units.

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Write a formula for the function obtained when the graph of f ( x ) = | x | is shifted down 3 units and to the right 1 unit.

g ( x ) = | x - 1 | 3

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Write a formula for the function obtained when the graph of f ( x ) = 1 x is shifted down 4 units and to the right 3 units.

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Write a formula for the function obtained when the graph of f ( x ) = 1 x 2 is shifted up 2 units and to the left 4 units.

g ( x ) = 1 ( x + 4 ) 2 + 2

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For the following exercises, describe how the graph of the function is a transformation of the graph of the original function f .

y = f ( x + 43 )

The graph of f ( x + 43 ) is a horizontal shift to the left 43 units of the graph of f .

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y = f ( x 4 )

The graph of f ( x - 4 ) is a horizontal shift to the right 4 units of the graph of f .

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y = f ( x ) + 8

The graph of f ( x ) + 8 is a vertical shift up 8 units of the graph of f .

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y = f ( x ) 7

The graph of f ( x ) 7 is a vertical shift down 7 units of the graph of f .

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y = f ( x + 4 ) 1

The graph of f ( x + 4 ) 1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f .

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For the following exercises, determine the interval(s) on which the function is increasing and decreasing.

f ( x ) = 4 ( x + 1 ) 2 5

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g ( x ) = 5 ( x + 3 ) 2 2

decreasing on ( , 3 ) and increasing on ( 3 , )

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k ( x ) = 3 x 1

decreasing on ( 0 , )

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Graphical

For the following exercises, use the graph of f ( x ) = 2 x shown in [link] to sketch a graph of each transformation of f ( x ) .

Graph of f(x).

For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions.

f ( t ) = ( t + 1 ) 2 3

Graph of f(t).
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h ( x ) = | x 1 | + 4

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k ( x ) = ( x 2 ) 3 1

Graph of k(x).
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Numeric

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

x −2 −1 0 1 2
f ( x ) −2 −1 −3 1 2
x −1 0 1 2 3
g ( x ) −2 −1 −3 1 2
x −2 −1 0 1 2
h ( x ) −1 0 −2 2 3

g ( x ) = f ( x - 1 ) , h ( x ) = f ( x ) + 1

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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