3.6 Review and Summary
Additional Information
Access this online video resource for additional instruction and practice with transformation of functions.
Key Equations
| Vertical shift | [latex]g\left(x\right)=f\left(x\right)+k\text{}[/latex] (up for [latex]\text{}k>0[/latex]) |
| Horizontal shift | [latex]g\left(x\right)=f\left(x-h\right)[/latex] (right for [latex]\text{}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)\text{}[/latex] ([latex]a>0[/latex]) |
| Vertical compression | [latex]g\left(x\right)=af\left(x\right)\text{}[/latex] ([latex]0 < a < 1[/latex]) |
| Horizontal stretch | [latex]g\left(x\right)=f\left(bx\right)[/latex] ([latex]0 < b < 1[/latex]) |
| Horizontal compression | [latex]g\left(x\right)=f\left(bx\right)\text{}[/latex] ([latex]b > 1[/latex]) |
Key Terms
- Even Function – a function whose graph is unchanged by horizontal reflection, [latex]\text{}f\left(x\right)=f\left(-x\right)\text{}[/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]\text{}b>1[/latex].
- Horizontal Reflection – a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]\text{}-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]
- 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]\text{}-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]\text{}a>1[/latex].
Key Concepts
- A function can be shifted vertically by adding a constant to the output. See 3.2 Graphing Functions Using Vertical and Horizontal Shifts.
- A function can be shifted horizontally by adding a constant to the input. See 3.2 Graphing Functions Using Vertical and Horizontal Shifts.
- Relating the shift to the context of a problem makes it possible to compare and interpret vertical and horizontal shifts. See 3.2 Graphing Functions Using Vertical and Horizontal Shifts.
- Vertical and horizontal shifts are often combined. See 3.2 Graphing Functions Using Vertical and Horizontal Shifts.
- A vertical reflection reflects a graph about the [latex]\text{}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. See 3.3 Graphing Functions Using Reflections about the Axes.
- A function presented in tabular form can also be reflected by multiplying the values in the input and output rows or columns accordingly. See 3.3 Graphing Functions Using Reflections about the Axes.
- A function presented as an equation can be reflected by applying transformations one at a time. See 3.3 Graphing Functions Using Reflections about the Axes.
- 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]\text{}f\left(x\right)=f\left(-x\right)[/latex].
- Odd functions satisfy the condition [latex]\text{}f\left(x\right)=-f\left(-x\right)[/latex].
- A function can be odd, even, or neither. See 3.4 Determining Even and Odd Functions.
- A function can be compressed or stretched vertically by multiplying the output by a constant. See 3.5 Graphing Functions Using Stretches and Compressions.
- A function can be compressed or stretched horizontally by multiplying the input by a constant. See 3.5 Graphing Functions Using Stretches and Compressions.
- 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 3.5 Graphing Functions Using Stretches and Compressions.
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