MathematicsPreCalculus Mathematics at Nebraska

SubsectionReflection About y-axis

Let $f(x)=(x+1)^2+1\text{,}$ and set

Compared to $f(x)\text{,}$ the graph of $h(x)$ is flipped, or reflected, about the $y$-axis as shown below.

$x$	$f(x)$	$h(x)=f(-x)$
$-2$	$2$	$10$
$-1$	$1$	$5$
$0$	$2$	$2$
$1$	$5$	$1$
$2$	$10$	$2$

Looking at the graph and table above, we notice a relationship between the outputs of $f(x)$ and $f(-x)$ in this particular case, and as it so happens, this relationship holds in general.

Notice that if you take the column of output values for $f(x)$ and flip them upside down you get the outputs of $f(-x)\text{.}$ This makes sense as this is saying, for example,

as $h(x)=f(-x)\text{.}$ In general, we have the following definition:

There are some functions which do not change when we apply a reflection about the $y$-axis. The most common example of this is the function $y=x^2\text{.}$ Looking at a graph of this function, it is easy to see that when you reflect it across the $y$-axis it remains unchanged. These sorts of functions have a particular name:

To help remember the definition of an even function, notice that the example of an even function we gave was of $y=x^2\text{.}$ Other examples are $y= x^4 \text{,}$ $y=x^6 \text{,}$ $y=x^8\text{,}$ etc. Notice that the exponent of each of these functions is an even number.

SubsectionReflection About x-axis

Compared to $y=x^2\text{,}$ the graph of $h(x) = -x^2$ is flipped, or reflected, about the $x$-axis. The $y$-coordinate of each point on the graph of $y = x^2$ is replaced by its additive inverse.

$x$	$y=x^2$	$h(x)=-x^2$
$-2$	$4$	$-4$
$-1$	$1$	$-1$
$0$	$0$	$0$
$1$	$1$	$-1$
$2$	$4$	$-4$

In general, we have the following definition:

We now know that a function satisfying $f(x)=f(-x)$ is an even function. While there do not exist any functions satisfying $f(x)=-f(x)\text{,}$ we do have another type of symmetry to discuss here.

Example267

Consider the function $f(x)=x^3 \text{.}$ If we reflect $f$ about the $y$-axis, we get the function $f(-x)$ as shown below:

If we now reflect the function $f(-x)$ about the $x$ axis, we can see from the above picture that we end up back with the original function. As an equation, this says

$$\begin{equation*} f(x)=-f(-x). \end{equation*}$$

This type of function is said to have symmetry about the origin, or called an odd function.

To help remember the definition of an odd function, we have a similar strategy as for even functions. Some examples of odd functions are $y=x^3\text{,}$ $y=x^5\text{,}$ $y=x^7 \text{,}$ etc. Each of these examples have exponents which are odd numbers, and they are odd functions.