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Finding a triple transformation of a graph

Use the graph of f ( x ) in [link] to sketch a graph of k ( x ) = f ( 1 2 x + 1 ) 3.

Graph of a half-circle.

To simplify, let’s start by factoring out the inside of the function.

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

By factoring the inside, we can first horizontally stretch by 2, as indicated by the 1 2 on the inside of the function. Remember that twice the size of 0 is still 0, so the point (0,2) remains at (0,2) while the point (2,0) will stretch to (4,0). See [link] .

Graph of a vertically stretch half-circle.

Next, we horizontally shift left by 2 units, as indicated by x + 2. See [link] .

Graph of a vertically stretch and translated half-circle.

Last, we vertically shift down by 3 to complete our sketch, as indicated by the 3 on the outside of the function. See [link] .

Graph of a vertically stretch and translated half-circle.
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Access this online resource for additional instruction and practice with transformation of functions.

Key equations

Vertical shift g ( x ) = f ( x ) + k (up for k > 0 )
Horizontal shift g ( x ) = f ( x h ) (right for h > 0 )
Vertical reflection g ( x ) = f ( x )
Horizontal reflection g ( x ) = f ( x )
Vertical stretch g ( x ) = a f ( x ) ( a > 0 )
Vertical compression g ( x ) = a f ( x ) ( 0 < a < 1 )
Horizontal stretch g ( x ) = f ( b x ) ( 0 < b < 1 )
Horizontal compression. g ( x ) = f ( b x ) ( b > 1 )

Key concepts

  • A function can be shifted vertically by adding a constant to the output. See [link] and [link] .
  • A function can be shifted horizontally by adding a constant to the input. See [link] , [link] , and [link] .
  • Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See [link] .
  • Vertical and horizontal shifts are often combined. See [link] and [link] .
  • A vertical reflection reflects a graph about the x - axis. A graph can be reflected vertically by multiplying the output by –1.
  • A horizontal reflection reflects a graph about the y - axis. A graph can be reflected horizontally by multiplying the input by –1.
  • A graph can be reflected both vertically and horizontally. The order in which the reflections are applied does not affect the final graph. See [link] .
  • A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See [link] .
  • A function presented as an equation can be reflected by applying transformations one at a time. See [link] .
  • Even functions are symmetric about the y - axis, whereas odd functions are symmetric about the origin.
  • Even functions satisfy the condition f ( x ) = f ( x ) .
  • Odd functions satisfy the condition f ( x ) = f ( x ) .
  • A function can be odd, even, or neither. See [link] .
  • A function can be compressed or stretched vertically by multiplying the output by a constant. See [link] , [link] , and [link] .
  • A function can be compressed or stretched horizontally by multiplying the input by a constant. See [link] , [link] , and [link] .
  • The order in which different transformations are applied does affect the final function. Both vertical and horizontal transformations must be applied in the order given. However, a vertical transformation may be combined with a horizontal transformation in any order. See [link] and [link] .

Section exercises

Verbal

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?

A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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