# 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 $\text{}f\left(x\right)$, a new function $\text{}g\left(x\right)=-f\left(x\right)\text{}$ is a vertical reflection of the function $\text{}f\left(x\right)\text{}$, sometimes called a reflection about (or over, or through) the x-axis.

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

## How To

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

1. Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the x-axis.
2. 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

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

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

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

1. $\text{}g\left(x\right)=-f\left(x\right)$
2. $\text{}h\left(x\right)=f\left(-x\right)$
 $x$ $f\left(x\right)$ 2 4 6 8 1 3 7 11

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

1. $g\left(x\right)=-f\left(x\right)$
2. $h\left(x\right)=f\left(-x\right)$
 $x$ $f\left(x\right)$ −2 0 2 4 5 10 15 20

## Example 3: Applying a Learning Model Equation

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

### Analysis

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

## Example 4: Graphing Functions

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