# How to reflect a graph through the x-axis, y-axis or Origin?

By Murray Bourne, 30 Jun 2011

This mail came in from reader Stuart recently:

Can you explain the principles of a graph involving

y= βf(x) being a reflection of the graphy=f(x) in thex-axis and the graph ofy=f(βx) a reflection of the graphy=f(x) in they-axis?Thanks

## My reply

Hello Stuart

Let's see what this means via an example.

Let *f*(*x*) = 3*x* + 2

If you are not sure what it looks like, you can graph it using this graphing facility.

You'll see it is a straight line, slope 3 (which is positive, i.e. going uphill as we go left to right) and *y*-intercept 2.

Now let's consider β*f*(*x*).

This gives us

β*f*(*x*)* =* β3*x* β 2

Our new line has negative slope (it goes down as you scan from left to right) and goes through β2 on the *y*-axis.

When you graph the 2 lines on the same axes, it looks like this:

Note that if you reflect the blue graph (*y* = 3*x* + 2) in the *x*-axis, you get the green graph (*y* = β3*x* β 2) (as shown by the red arrows).

What we've done is to take every *y*-value and turn them upside down (this is the effect of the **minus** out the front).

## Now for *f*(β*x*)

Similarly, let's do *f*(β*x*).

Since *f*(*x*) = 3*x* + 2, then

*f*(*βx*) = β3*x* + 2 (replace every "*x*" with a "*βx*").

Now, graphing those on the same axes, we have:

Note that the effect of the "minus" in *f*(*βx*) is to reflect the blue original line (*y* = 3*x* + 2) in the *y*-axis, and we get the green line, which is (*y* = β3*x* + 2). The green line also goes through 2 on the *y*-axis.

## Further Example

Here's an example using a cubic graph.

**Blue graph:** *f*(*x*) = *x*^{3} β 3*x*^{2} + *x* β 2

**Reflection in x-axis (green):** β*f*(*x*) = β*x*^{3} + 3*x*^{2} β *x* + 2

Now to reflect in the y-axis.

**Blue graph:** *f*(*x*) = *x*^{3} β 3*x*^{2} + *x* β 2

**Reflection in y-axis (green): ***f*(*βx*) = *βx*^{3} β 3*x*^{2} β *x* β 2

## Even and Odd Functions

We really should mention **even and odd functions** before leaving this topic.

For each of my examples above, the reflections in either the *x*- or *y*-axis produced a graph that was **different**. But sometimes, the reflection is the same as the original graph. We say the reflection "maps on to" the original.

**Even Functions**

An **even **function has the property *f*(*βx*) = *f*(*x*). That is, if we reflect an even function in the *y*-axis, it will look exactly like the original.

An example of an even function is *f*(*x*) = *x*^{4} β 29*x*^{2} + 100

The above even function is equivalent to:

*f*(*x*) = (*x + *5)(*x * + 2)(*x *β 2)(*x β* 5)

Note if we reflect the graph in the *y*-axis, we get the same graph (or we could say it "maps onto" itself).

**Odd Functions**

An **odd **function has the property *f*(*βx*) = *βf*(*x*).

This time, if we reflect our function in **both **the *x*-axis and *y*-axis, and if it looks exactly like the original, then we have an odd function.

This kind of symmetry is called** origin symmetry**. An odd function either passes through the origin (0, 0) or is reflected through the origin.

An example of an odd function is *f*(*x*) = *x*^{3} β 9*x*

The above odd function is equivalent to:

*f*(*x*) = *x*(*x + *3)(*x *β 3)

Note if we reflect the graph in the *x*-axis, then the *y*-axis, we get the same graph.

## More examples of Even and Odd functions

There some more examples on this page: Even and Odd Functions

Knowing about even and odd functions is very helpful when studying Fourier Series.

I hope that all makes sense, Stuart.

See the 8 Comments below.