# 3.5 Transformation of functions  (Page 10/21)

 Page 10 / 21

## Finding a triple transformation of a graph

Use the graph of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ in [link] to sketch a graph of $\text{\hspace{0.17em}}k\left(x\right)=f\left(\frac{1}{2}x+1\right)-3.$

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

$f\left(\frac{1}{2}x+1\right)-3=f\left(\frac{1}{2}\left(x+2\right)\right)-3$

By factoring the inside, we can first horizontally stretch by 2, as indicated by the $\text{\hspace{0.17em}}\frac{1}{2}\text{\hspace{0.17em}}$ 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] .

Next, we horizontally shift left by 2 units, as indicated by $\text{\hspace{0.17em}}x+2.\text{\hspace{0.17em}}$ See [link] .

Last, we vertically shift down by 3 to complete our sketch, as indicated by the $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ on the outside of the function. See [link] .

Access this online resource for additional instruction and practice with transformation of functions.

## Key equations

 Vertical shift $g\left(x\right)=f\left(x\right)+k\text{\hspace{0.17em}}$ (up for $\text{\hspace{0.17em}}k>0$ ) Horizontal shift $g\left(x\right)=f\left(x-h\right)$ (right for $\text{\hspace{0.17em}}h>0$ ) Vertical reflection $g\left(x\right)=-f\left(x\right)$ Horizontal reflection $g\left(x\right)=f\left(-x\right)$ Vertical stretch $g\left(x\right)=af\left(x\right)\text{\hspace{0.17em}}$ ( $a>0$ ) Vertical compression $g\left(x\right)=af\left(x\right)\text{\hspace{0.17em}}$ $\left(0 Horizontal stretch $g\left(x\right)=f\left(bx\right)$ $\left(0 Horizontal compression. $g\left(x\right)=f\left(bx\right)\text{\hspace{0.17em}}$ ( $b>1$ )

## Key concepts

• A function can be shifted vertically by adding a constant to the output. See [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 $\text{\hspace{0.17em}}x\text{-}$ axis. A graph can be reflected vertically by multiplying the output by –1.
• A horizontal reflection reflects a graph about the $y\text{-}$ 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 $y\text{-}$ axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition $\text{\hspace{0.17em}}f\left(x\right)=f\left(-x\right).$
• Odd functions satisfy the condition $\text{\hspace{0.17em}}f\left(x\right)=-f\left(-x\right).$
• 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] .

## 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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