## Key Equations

 Vertical shift $g\left(x\right)=f\left(x\right)+k$ (up for $k>0$ ) Horizontal shift $g\left(x\right)=f\left(x-h\right)$ (right for $h>0$ ) Vertical reflection $g\left(x\right)=-f\left(x\right)$ Horizontal reflection $g\left(x\right)=f\left(-x\right)$ Vertical stretch $g\left(x\right)=af\left(x\right)$ ( $a>0$) Vertical compression $g\left(x\right)=af\left(x\right)$ $\left(01$ )

## Key Concepts

• A function can be shifted vertically by adding a constant to the output.
• A function can be shifted horizontally by adding a constant to the input.
• Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts.
• Vertical and horizontal shifts are often combined.
• A vertical reflection reflects a graph about the $x\text{-}$ 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.
• A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly.
• A function presented as an equation can be reflected by applying transformations one at a time.
• Even functions are symmetric about the $y\text{-}$ axis, whereas odd functions are symmetric about the origin.
• Even functions satisfy the condition $f\left(x\right)=f\left(-x\right)$.
• Odd functions satisfy the condition $f\left(x\right)=-f\left(-x\right)$.
• A function can be odd, even, or neither.
• A function can be compressed or stretched vertically by multiplying the output by a constant.
• A function can be compressed or stretched horizontally by multiplying the input by a constant.
• 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.

## Glossary

even function
a function whose graph is unchanged by horizontal reflection, $f\left(x\right)=f\left(-x\right)$, and is symmetric about the $y\text{-}$ axis
horizontal compression
a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant $b>1$
horizontal reflection
a transformation that reflects a function’s graph across the y-axis by multiplying the input by $-1$
horizontal shift
a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
horizontal stretch
a transformation that stretches a function’s graph horizontally by multiplying the input by a constant $0<b<1$
odd function
a function whose graph is unchanged by combined horizontal and vertical reflection, $f\left(x\right)=-f\left(-x\right)$, and is symmetric about the origin
vertical compression
a function transformation that compresses the function’s graph vertically by multiplying the output by a constant $0<a<1$
vertical reflection
a transformation that reflects a function’s graph across the x-axis by multiplying the output by $-1$
vertical shift
a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
vertical stretch
a transformation that stretches a function’s graph vertically by multiplying the output by a constant $a>1$