Try reviewing this fundamentals firstTransformations that functions: Horizontal translationsTransformations that functions: vertical translations

Still don't gain it?Review these simple concepts…Transformations of functions: Horizontal translationsTransformations of functions: upright translationsNope, I got it.

You are watching: How do you reflect across the x axis

Reflection throughout the X-Axis

Before we acquire into reflections throughout the y-axis, make certain you've refreshed her memory on just how to do straightforward vertical and also horizontal translations.

How to Reflect over X-Axis:

One of the most straightforward transformations you deserve to make with straightforward functions is come reflect it throughout the x-axis or another horizontal axis. In a potential check question, this have the right to be grammar in many different ways, so make certain you acknowledge the following terms as just another means of saying "perform a reflection throughout the x-axis":

1) Graph y=−f(x)y = -f(x)y=−f(x)

2) Graph −f(x)-f(x)−f(x)

3) Reflect over xxx axis

In bespeak to execute this, the process is exceptionally simple: For any function, no issue how facility it is, just pick out easy-to-determine coordinates, divide the y-coordinate through (-1), and then re-plot those coordinates. That's it!

The best method to practice illustration reflections throughout the y-axis is to do an instance problem:


Given the graph of y=f(x)y = f(x)y=f(x) as shown, lay out y=−f(x)y = -f(x)y=−f(x).

Remember, the only step we need to do before plotting the −f(x)-f(x)−f(x) have fun is merely divide the y-coordinates that easy-to-determine points on our graph above by (-1). Once we speak "easy-to-determine points" what this refers to is just points for which you recognize the x and also y values exactly. Don't pick points whereby you need to estimate values, together this makes the problem unnecessarily hard. Below are several images to help you visualize exactly how to deal with this problem.

Step 1: understand that we're reflecting throughout the x-axis

Since us were asked to plot the –f(x)f(x)f(x) reflection, is it very important the you recognize this means we room being asked come plot the reflection over the x-axis. When illustration reflections across the xxx and yyy axis, that is really easy come get confused by several of the notations. So, make certain you take it a moment prior to solving any kind of reflection problem to check you understand what you're being asked to do.

Step 2: determine easy-to-determine points

Remember, choose some clues (3 is generally enough) the are straightforward to choose out, meaning you know exactly what the x and y worths are. In this case, let's choose (-2 ,-3), (-1 ,0), and also (0,3).

Pick 3 points and graph

Step 3: divide these points by (-1)

While the xxx values continue to be the same, all we need to do is divide the yyy worths by (-1)!

divide y values by -1
Plot new points after splitting y worths by -1

And that's it! Simple, right?

What is the Axis the Symmetry:

In some cases, you will be asked to perform horizontal reflections across an axis of symmetry that isn't the x-axis. But prior to we go into how to resolve this, it's crucial to know what we median by "axis the symmetry". The axis of symmetry is simply the horizontal line that we space performing the have fun across. It have the right to be the x-axis, or any kind of horizontal line with the equation yyy = constant, favor yyy = 2, yyy = -16, etc.

How to discover the Axis of Symmetry:

Finding the axis that symmetry, favor plotting the reflections themselves, is likewise a straightforward process. In this case, every we have to do is choose the same point ~ above both the function and that reflection, count the distance between them, division that by 2, and count that distance away from one of the graphs. This is because, by it's definition, an axis of the contrary is exactly in the center of the role and that reflection.

The best means to practice finding the axis of the contrary is to do an example problem.


Find the axis that symmetry because that the two functions displayed in the pictures below.

Graph A come reflect through x axis

Again, all we have to do to solve this problem is to choose the same allude on both functions, counting the distance between them, divide by 2, and then include that street to among our functions. Let's pick the origin suggest for these functions, as it is the easiest allude to deal with.

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Now, by count the distance between these 2 points, you should get the price of 2 units. The last action is to division this value by 2, offering us 1. Now we recognize that our axis of the opposite is exactly one unit below the peak function's origin or over the bottom features origin. Looking in ~ the graph, this offers us yyy = 5 together our axis the symmetry! Let's take a look in ~ what this would look like if there to be an actual heat there:

And that's all there is come it! You might learn further on just how to graph transformations of trigonometric functions and also how to recognize trigonometric attributes from their graphs in other sections.