Lesson Objectives
• Learn about symmetry with respect to the x-axis, y-axis, and origin
• Learn how to determine if a function is even
• Learn how to determine if a function is odd

## How to Determine if a Function is Even, Odd, or Neither

Graphs of equations can have symmetry with respect to the x-axis, the y-axis, or the origin.

### Symmetric with Respect to the x-axis

When we have symmetry with respect to the x-axis, we could fold the coordinate plane along the x-axis and the part of the graph above the x-axis would coincide with the part of the graph below the x-axis. The graph of an equation is symmetric with respect to the x-axis if the replacement of y with -y results in an equivalent equation. In other words, if (x, y) is on the graph, then (x, -y) is also on the graph. Looking at the graph above, we can see symmetry with respect to the x-axis. If we take any point on the graph (x, y), and reflect across the x-axis, we will end up with another point on the graph (x, -y). For example, reflecting the point (9, 3) across the x-axis, gives us the point (9, -3). Since this graph does not pass the vertical line test, it is not a function.

### Symmetric with Respect to the y-axis

When the graph of an equation is symmetric with respect to the y-axis, we could fold the coordinate plane along the y-axis and the part of the graph on the right of the y-axis would coincide with the part of the graph on the left of the y-axis. The graph of an equation is symmetric with respect to the y-axis if the replacement of x with -x results in an equivalent equation. In other words, if (x, y) is on the graph, then (-x, y) is also on the graph. Looking at the graph above, we can see symmetry with respect to the y-axis. If we take any point on the graph (x, y), and reflect across the y-axis, we will end up with another point on the graph (-x, y). For example, reflecting the point (2, 4) across the y-axis, gives us the point (-2, 4).

### Even Functions

A function is even if it is symmetric with respect to the y-axis. Recall from above, that a function that is symmetric with respect to the y-axis will have the points (x, y) and (-x, y) on its graph. This leads to the following rule for even functions. A function f is called an even function if: $$f(x)=f(-x)$$ for all x in its domain. To determine if a function is even, we can plug in -x for x in our function and see if we get a match. Let's look at a few examples.
Example #1: Determine if the function is even. $$f(x)=-5x^2 - 3x + 1$$ Let's plug in a -x for x and see if f(-x) = f(x). $$f(-x)=-5(-x)^2 - 3(-x) + 1$$ $$f(-x)=-5x^2 + 3x + 1$$ As we can see, f(-x) is not equal to f(x): $$f(x)=-5x^2 - 3x + 1$$ $$f(-x)=-5x^2 + 3x + 1$$ Therefore, we can conclude this function is not even. Unless you are using graphing software, it's not very practical to use a graphical approach. We can use it when learning the concept to visually show this function is not symmetric with respect to the y-axis, and therefore, not an even function.
Desmos Link for More Detail Example #2: Determine if the function is even. $$f(x)=2x^4 + x^2 - 1$$ Let's plug in a -x for x and see if f(-x) = f(x). $$f(-x)=2(-x)^4 + (-x)^2 - 1$$ $$f(-x)=2x^4 + x^2 - 1$$ As we can see, f(x) is equal to f(-x): $$f(x)=2x^4 + x^2 - 1$$ $$f(-x)=2x^4 + x^2 - 1$$ Therefore, we can conclude this function is even. Again, we can always check using a graphical approach.
Desmos Link for More Detail ### Symmetric with Respect to the Origin

A different type of symmetry happens when a graph can be rotated 180° about the origin, with the result coinciding exactly with the original graph. This type of symmetry is known as being symmetric with respect to the origin. When a graph is symmetric with respect to the origin, if the point (x, y) is on the graph, then the point (-x, -y) is also on the graph. Looking at the graph above, we can see symmetry with respect to the origin. Reflecting a point (x, y) across the origin will give you (-x, -y). For example, reflecting the point (2, 8) across the origin, gives us the point (-2, -8).
Origin symmetry is a lot harder to think about visually versus symmetry with respect to the x-axis or y-axis. Some students find it easier to think about origin symmetry as reflecting across the x-axis and then the y-axis and obtaining the same graph.

### Odd Functions

A function is odd if it is symmetric with respect to the origin. Recall from above, that a function that is symmetric with respect to the origin will have the points (x, y) and (-x, -y) on its graph. This leads to the following rule for odd functions. A function f is called an odd function if: $$f(-x)=-f(x)$$ for all x in its domain. To determine if a function is odd, we can plug in -x for x in our function and see if the result is the same as -f(x). Let's look at an example.
Example #3: Determine if the function is odd. $$f(x)=x^3 - 7x$$ Let's plug in and see if -f(x) = f(-x). $$f(-x)=(-x)^3 - 7(-x)$$ $$f(-x)=-x^3 + 7x$$ Now, let's find -f(x) and see if they match. $$-f(x)=-1 \cdot (x^3 - 7x)$$ $$-f(x)=-x^3 + 7x$$ As we can see, -f(x) is equal to f(-x). $$f(-x)=-x^3 + 7x$$ $$-f(x)=-x^3 + 7x$$ Therefore, we can conclude this function is odd. Again, unless you are using graphing software, it's not very practical to use a graphical approach. Once again, we will include a graph here to visually show this function is symmetric with respect to the origin, and therefore, an odd function.
Desmos Link for More Detail #### Skills Check:

Example #1

Determine if even, odd, or neither. $$f(x)=-4x^3 - \frac{1}{2}x + 2$$

A
Even
B
Odd
C
Neither

Example #2

Determine if even, odd, or neither. $$f(x)=5|x| + x^2$$

A
Even
B
Odd
C
Neither

Example #3

Determine if even, odd, or neither. $$f(x)=-x^3 - 3\sqrt{x}$$

A
Even
B
Odd
C
Neither         