Reflect Function About y-Axis: f(-x)

Reflecting a Function About the $y$-Axis

In the previous lesson, we saw the function transformation of reflecting a function about the $x$-axis, where $y=-f(x)$. In this lesson, we're going to talk about reflecting a function over the other axis: the $y$-axis.

The $y$-axis is the vertical axis, so if we were to reflect a function over the $y$-axis, we would do a horizontal flip. That means that the $y$-values would stay the same, but the $x$-values will flip.

$$y=f(-x)$$

It might be helpful to talk through an example.

Example

Say we have the quadratic function $y=(x-3)^2$. This parabola looks like:

If we consider the $y$-axis our mirror, we can do a horizontal reflection by negating all of the $x$-values. Essentially, we are just flipping this parabola into quadrant II. The graph in purple represents our reflected function.

Horizontal Reflection

The next lesson in function transformations is when you can horizontally reflect an image across the $y$-axis.

When we are doing a horizontal reflection, the $y$ values stay the same, as the $y$-axis is the line of reflection. Think of the $y$-axis functioning as a mirror! The $x$ values are the ones that will change, and they will be negated, meaning you will flip the sign of the $x$ values.

$$f(-x)$$

Let's walk through an example.

Example

Let's say you were given the function $f(x)=\left(x+1\right)^{3}$. The graph of this function looks like:

If we wanted to reflect this graph over the $y$-axis, we would keep all of the $y$-coordinates the same, but the signs on the $x$-values would be flipped. The transformed function equation would look like:

$$f(x)=\left(-x+1\right)^{3}$$

The graph of this function is in green, while the graph of the original function is in purple.

As you can see from the image, it looks like the $y$-axis is the line of reflection for the functions! Either side of the $y$-axis is a mirror image.

Reflect a Line oVer the Y-Axis

by Brian McLogan

In this video by Brian McLogan, you will learn how to reflect a line over the y-axis, which you can apply to other functions.