Reflecting Functions

Reflections are quite simple. When you reflect something, you are basically flipping it across a given line. There are many types of reflections, depending across the line you are reflecting across of. However, there are two types of basic reflections: about the x-axis, and about the y-axis.

Reflecting about the x-axis means you are rotating the function across the x-axis. This is done by multiplying the function by -1, so that you are left with –f(x).
For example,
The reflection of f(x)=x/2-3 will be the same as multiplying f(x)=x/2-3 by -1, which is –f(x)=-x/2+3
Graphically, it would look like this:

f(x)=x/2-3

–f(x)=-x/2+3Another example:

f(x)=x^2-3 -f(x)=-(x^2-3)

To understand this type of reflection better, think of it the following way. By reflecting about the x-axis, you are multiplying each value of y by -1. That means that the reflection about the x-axis of the point (1,1) would be (1,-1). Using variables, this is the same as saying that the reflection about the x-axis of the point (x,y) is (x,-y). You can prove this using the graph above. Each value of y gets multiplied by -1 while x stays the same. Therefore, if the original function is f(x), the reflected function about the x-axis will be -f(x)

Reflecting about the y-axis means you are rotating the function across the y-axis. This is done by replacing x by -x, so that you are left with f(-x).
For example,
The reflection of f(x)=x/2-3 will be the same as solving for f(-x), which is f(-x)=-x/2-3
Graphically, it would look like this:

f(x)=x/2-3

f(-x)=-x/2-3

Another example:

f(x)=(x-3)^2 f(-x)=(-x-3)^2

To understand this type of reflection better, think of it the following way. By reflecting about the y-axis, you are multiplying each value of x by -1. That means that the reflection about the y-axis of the point (3,0) would be (-3,0). Using variables, this is the same as saying that the reflection about the y-axis of the point (x,y) is (-x,y). You can prove this using the graph above. Each value of x gets multiplied by -1 while y stays the same. Therefore, if the original function is f(x), the reflected function about the y-axis will be f(-x)

Review:
If f(x) is a function, then
-f(x) will give you its reflection about the x-axis, and
f(-x) will give you its reflection about the y-axis

The following link contains other examples: http://www.themathpage.com/aPreCalc/reflections.htm

If you would like some help on other types of reflections (about the line y=x, y=-x, etc), please let me know.