Summary: Transformations of Functions

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

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