## Key Equations

Vertical shift | [latex]g\left(x\right)=f\left(x\right)+k[/latex] (up for [latex]k>0[/latex] ) |

Horizontal shift | [latex]g\left(x\right)=f\left(x-h\right)[/latex] (right for [latex]h>0[/latex] ) |

Vertical reflection | [latex]g\left(x\right)=-f\left(x\right)[/latex] |

Horizontal reflection | [latex]g\left(x\right)=f\left(-x\right)[/latex] |

Vertical stretch | [latex]g\left(x\right)=af\left(x\right)[/latex] ( [latex]a>0[/latex]) |

Vertical compression | [latex]g\left(x\right)=af\left(x\right)[/latex] [latex]\left(0<a<1\right)[/latex] |

Horizontal stretch | [latex]g\left(x\right)=f\left(bx\right)[/latex] [latex]\left(0<b<1\right)[/latex] |

Horizontal compression | [latex]g\left(x\right)=f\left(bx\right)[/latex] ( [latex]b>1[/latex] ) |

## 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 [latex]x\text{-}[/latex] axis. A graph can be reflected vertically by multiplying the output by –1.
- A horizontal reflection reflects a graph about the [latex]y\text{-}[/latex] 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 [latex]y\text{-}[/latex] axis, whereas odd functions are symmetric about the origin.
- Even functions satisfy the condition [latex]f\left(x\right)=f\left(-x\right)[/latex].
- Odd functions satisfy the condition [latex]f\left(x\right)=-f\left(-x\right)[/latex].
- 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, [latex]f\left(x\right)=f\left(-x\right)[/latex], and is symmetric about the [latex]y\text{-}[/latex] axis

- horizontal compression
- a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant [latex]b>1[/latex]

- horizontal reflection
- a transformation that reflects a function’s graph across the
*y*-axis by multiplying the input by [latex]-1[/latex]

- 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 [latex]0<b<1[/latex]

- odd function
- a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\left(x\right)=-f\left(-x\right)[/latex], 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 [latex]0<a<1[/latex]

- vertical reflection
- a transformation that reflects a function’s graph across the
*x*-axis by multiplying the output by [latex]-1[/latex]

- 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 [latex]a>1[/latex]