Mencerminkan tentang x-aksis

Background

We've seen a couple of different function transformations already, including vertical shifts and horizontal shifts. This lesson will cover a reflection over the $x$-axis.

Reflect a Function About the $x$-Axis

To reflect a function about the $x$-axis, we want to negate all of the $y$ values. We do that because the horizontal position stays the same, so the $x$-coordinate will stay the same. What changes is the vertical position, so we just switch the sign on the $y$-coordinate.

$$y = -f(x)$$

Let's walk through an example.

Example

Say we are given a triangle with the points, $(0,1)$, $(4,6)$, and $(8,1)$.

To reflect this triangle, we will negate all of the $y$-coordiantes.

$$\begin{align} (0,1) \rightarrow (0,-1) \\ (4,6) \rightarrow (4,-6) \\ (8,1) \rightarrow (8,-1) \\ \end{align}$$

Here's a graph of the new reflected triangle:

Vertical Reflection

The next type of function transformation is the vertical reflection!

The vertical reflection allows us to flip an image over the $x$-axis. The horizontal $x$-axis functions as a mirror upon which you can negative the $y$ values, but keep the $x$ values the same.

$$-f(x)$$

Let's walk through an example.

Example

Given the function $y=3x^{4\ }-2$, reflect the function vertically, over the $x$-axis.

Here is the graph of the original function, $y=3x^{4\ }-2$:

As we saw earlier, if we negative the $y$ value, we can rewrite the function equation as: $-y=3x^{4}-2$. The graph of this vertically reflected graph looks like:

Let's take a look at both functions on the same graph!

The graph in red is the original function, and the graph in blue is the vertically reflected function! You can think of the graph in blue as a mirror image.

Looking Further

When you learn further math, you will learn that reflections are a type of linear transformation that can be represented using matrices.

Reflect a Line Segment Over the X-Axis

by Brian McLogan

In this lesson by Brian McLogan, you will learn how to reflect a line segment over the $x$-axis.