Module 2: Reflections and Symmetry

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Section III: Graph Transformations

Module 2: Reflections and Symmetry

REFLECTIONS

We have studied how adding or subtracting constants to the “inside” or “outside” of a function affects its graph. We will now begin to study what happens when we multiply the “inside” or “outside” of a function by constants. Before we consider what happens when we multiply by any constant, we shall first focus on the consequences of multiplying by .

example:Let the graph below define the function . On the same coordinate plane, graph if .

Let’s start with a few key points on the graph of and see where they end up on the graph of . Let’s use the points and .

Let’s start by evaluating :

Now we’ll make a table of values showing where these key points end up on the graph of .

x /
/ –2

Let’s plot the points we just found:

Finally, connect the dots with a piecewise linear function since was a piecewise linear function:

GENERALIZATION:
The graph of is a reflection of the graph of about the x-axis.

example:Let the graph below define the function . On the same coordinate plane, graph if .

Let’s start with a few key points on the graph of and see where they end up on the graph of . Let’s use the following points:

and .

Let’s start by evaluating :

Now we’ll make a table of values showing where these key points end up on the graph of .

x /

Let’s plot the points we just found, and graph by connecting the points with a piecewise linear function:

GENERALIZATION:
The graph of is a reflection of the graph of about the y-axis.

SYMMETRY

Some functions have graphs with special types of symmetries, and we can use the reflections we just studied to analyze these symmetries.

symmetry about the y-axis:

example:The graph of is given below. How does it compare with the graph of ?

It should be clear that if we reflect the graph of about the y-axis, we obtain the exact same graph! This can be represented algebraically with the statement .

We say that graphs like have symmetry about the y-axis and call functions with this type of symmetry even functions. (It helps me to remember that is an example of an even function since the power on x (i.e., 2) is an even number.)

A function f is even if its graph is symmetric about the y-axis, (which means that if the graph of f isn’t changed if it is reflected about the y-axis). An algebraic test to determine if a function is even is given below:
A function f is even if, for all x in its domain, .
Notice that the test is an algebraic representation of the statement “a function is even if reflection about the y-axis does not change the graph.”

symmetry about the origin:

example:The graph of is given below. How does the graph of compare with the graph of ?

Notice that if you anchor the graph of at the origin and rotate it 180° in either direction, the graph ends up in the same place it started. We say that graphs with this sort of symmetry have symmetry about the origin.

We can also study this symmetry by considering reflections. Let’s reflect the graph of about both the x- and y-axes, i.e., let’s graph both (reflection about the y-axis) and (reflection about the x-axis):

The graph of .
(reflection of about y-axis) /
The graph of .
(reflection of about x-axis)

These graphs show us that when we reflect about the y-axis we obtain the same graph as we do when we reflect the graph about the x-axis. We can summarize this fact with the following algebraic statement: .

Functions with symmetry about the origin are called odd functions. (It helps me to remember that is an example of an odd function since the power on x (i.e., 3) is an odd number.)

A function is odd if its graph is symmetric about the origin, which means that if you rotate its graph 180° about the origin, you obtain the original graph. (Equivalently, a function is odd if reflection about the y-axis gives you the same graph as reflection about the x-axis.) An algebraic test to determine if a graph is odd is given below:
For all x in its domain, .
Notice that the test is an algebraic representation of the statement “a function is odd if reflection about the y-axis gives you the same graph as reflection about the x-axis.”

example:Perform the appropriate algebraic test to determine if the following functions are even, odd, or neither.

a.b.c.

SOLUTIONS:

It is important to notice that the algebraic tests for both even and odd symmetry start with making the input the opposite sign, so when we test for symmetry, we need to start with this step, simplify, and observe the result. We only need to perform one test and observe the result, rather than performing a test for both even and odd.

a.

Since , we can conclude that is an odd function.

b.

Since this is neither the original function nor the opposite of the original function, i.e., and , we see that is neither even nor odd.

c.

Since , we can conclude that is an even function.