# 3.3 Graphing Functions Using Reflections about the Axes

Another transformation that can be applied to a function is a reflection over the *x*– or *y*-axis. A **vertical reflection** reflects a graph vertically across the *x*-axis, while a **horizontal reflection** reflects a graph horizontally across the *y*-axis. The reflections are shown in Figure 3-9.

Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the *x*-axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the *y*-axis.

## Reflections

Given a function [latex]\text{}f\left(x\right)[/latex], a new function [latex]\text{}g\left(x\right)=-f\left(x\right)\text{}[/latex] is a vertical reflection of the function [latex]\text{}f\left(x\right)\text{}[/latex], sometimes called a reflection about (or over, or through) the *x*-axis.

Given a function [latex]\text{}f\left(x\right)\text{}[/latex], a new function [latex]\text{}g\left(x\right)=f\left(-x\right)\text{}[/latex] is a horizontal reflection of the function [latex]\text{}f\left(x\right)\text{}[/latex], sometimes called a reflection about the *y*-axis.

## How To

**Given a function, reflect the graph both vertically and horizontally. **

- Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the
*x*-axis. - Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the
*y*-axis.

## Example 1: Reflecting a Graph Horizontally and Vertically

- Reflect the graph of [latex]\text{}s\left(t\right)=\sqrt{t}[/latex] (a) vertically and (b) horizontally.
- Reflect the graph of [latex]\text{}f\left(x\right)=|x-1|[/latex] (a) vertically and (b) horizontally.

## Example 2: Reflecting a Tabular Function Horizontally and Vertically

1. A function [latex]\text{}f\left(x\right)\text{}[/latex] is given as Table 3. Create a table for the functions below.

- [latex]\text{}g\left(x\right)=-f\left(x\right)[/latex]
- [latex]\text{}h\left(x\right)=f\left(-x\right)[/latex]

[latex]x[/latex] |
2 | 4 | 6 | 8 |
---|---|---|---|---|

[latex]f\left(x\right)[/latex] |
1 | 3 | 7 | 11 |

2. A function [latex]\text{}f\left(x\right)\text{}[/latex] is given as Table 4. Create a table for the functions below.

- [latex]g\left(x\right)=-f\left(x\right)[/latex]
- [latex]h\left(x\right)=f\left(-x\right)[/latex]

[latex]x[/latex] |
−2 | 0 | 2 | 4 |
---|---|---|---|---|

[latex]f\left(x\right)[/latex] |
5 | 10 | 15 | 20 |

## Example 3: Applying a Learning Model Equation

A common model for learning has an equation similar to [latex]k\left(t\right)=-{2}^{-t}+1,\text{}[/latex] where [latex]k[/latex] is the percentage of mastery that can be achieved after [latex]t[/latex] practice sessions. This is a transformation of the function [latex]f\left(t\right)={2}^{t}[/latex] shown in Figure 3-10. Sketch a graph of [latex]k\left(t\right)[/latex].

*Analysis*

As a model for learning, this function would be limited to a domain of [latex]\text{}t\ge 0\text{}[/latex], with corresponding range [latex]\text{}\left[0,1\right)[/latex].

## Example 4: Graphing Functions

Given the toolkit function [latex]\text{}f\left(x\right)={x}^{2},\text{}[/latex] graph [latex]\text{}g\left(x\right)=-f\left(x\right)\text{}[/latex] and [latex]\text{}h\left(x\right)=f\left(-x\right)\text{}[/latex]. Take note of any surprising behaviour for these functions.

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