Reflections produce a mirror image of a function.  The reflection of a function can be performed along the $x$-axis, the $y$-axis, or any line.  For this section we will focus on the two axes and the line $y=x$. 

Vertical Reflections

A vertical reflection is a reflection across the $x$-axis, given by the equation:

$\displaystyle y=-f(x)$

In this general equation, all $y$ values are switched to their negative counterparts while the $x$ values remain the same. The result is that the curve becomes flipped over the $x$-axis.  As an example, let the original function be: 

$\displaystyle y = x^2$ 

The vertical reflection would then produce the equation:

$\displaystyle \begin{aligned} y &= -f(x)\\ & = -x^2 \end{aligned}$

Vertical reflection

The function $y=x^2$ is reflected over the $x$-axis.

Horizontal Reflections

A horizontal reflection is a reflection across the $y$-axis, given by the equation:

$\displaystyle y=f(-x)$

In this general equation, all $x$ values are switched to their negative counterparts while the y values remain the same. The result is that the curve becomes flipped over the $y$-axis.  Consider an example where the original function is:

$\displaystyle y = (x-2)^2$

Therefore the horizontal reflection produces the equation:

$\displaystyle \begin{aligned} y &= f(-x)\\ &= (-x-2)^2 \end{aligned}$

Horizontal reflection

The function $y=(x-2)^2$ is reflected over the $y$-axis.

Reflections Across a Line 

The third type of reflection is a reflection across a line.  Let's look at the case involving the line $y=x$.  This reflection has the effect of swapping the variables $x$and $y$, which is exactly like the case of an inverse function.  As an example, let the original function be:

$\displaystyle y = x^2$

The reflected equation, as reflected across the line $y=x$, would then be:

$y = \pm \sqrt{x}$

Reflection over $y=x$

The function $y=x^2$ is reflected over the line $y=x$.