5.3: Reflections & Even and Odd Functions

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Objectives

• Reflect a function across the $x$- and $y$-axis.
• Reflect and shift a function in the same problem.
• Students will be able to distinguish functions that are even, odd, or neither even or odd.
• Students will be able to use symmetries (even, odd) of a function.

Important Items

Definitions:

reflected across the $x-$axis, reflected across the $y-$axis, even function, odd function

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Lesson Guide

In this and the following sections, it's tempting to use a table to describe what transformations are doing. A huge downfall of this is that students may think that they always need a table in order to figure out what the graph of a transformation looks like, but aren't always given explicit functions for these tables to be useful. It's more beneficial to continue with the theme of 5.2 and focus on what's happening to points. Continue to write down sentences of the form "The height of the function $g$ at some input is somehow related to the height of $f$ at some input" to help students understand why formulas are the way they are.

Warm-Up

Have students do Problem 1.

Note: Many students are going to say they don't know what to do. Keep referring them back in their notes. They know what to do but are used to being spoon-fed the material.

Problem 2 is optional depending on how your time is going.

Reflect a graph across the $x$- and $y$-axis

Have students do Problem 3, which asks them to draw the reflected graph and create a table of new values. Make note of students approaching this question differently: some students start with drawing the graph and others start by trying to fill in the table. If you ask students to present, have them discuss their thought process. It may be the case that they looked at ordered pairs, which can help you motivate this process.

Help summarize problem 3 in the following way:

• "The height of $g$ at input $x$ is the negative of the height of $f$ at input $x$". You should write this on the board, and have your students to write it down as well. Notice that this sentence is quite literally what it means for one graph to be the reflection of the other across the $y$-axis. You should record the following observation:
  • If $g(x)=-f(x)$, then the graph of $g(x)$ is the graph of $f(x)$ reflected across the $x$-axis.

• "The height of $h$ at input $x$ is the height of $f$ at input $-x$." When they buy that, make the following observation:
  • If $g(x)=f(-x)$, then the graph of $g(x)$ is the graph of $f(x)$ reflected across the $y$-axis.

Have them work on Problems 4-5.

Reflect and shift a function in the same problem.

If you wish, do some examples similar to Problem 6.

Have students do Problem 6.

Recognize even and odd functions

Example:

* Some functions look exactly the same when we reflect across the $y$-axis. 
For example, consider $f(x)=x^2$: \[f(-x)=(-x)^2=(-1)^2(x)^2=(1)x^2=x^2=f(x).\] 
Graph and reflect $f(x)$ to illustrate this point.
* Other functions look exactly the same when we reflect across both the $x$- and $y$-axis. 
For example, consider $g(x)=x^3$: \[-g(-x)=-(-x)^3=-(-1)^3(x)^3=-(-1)x^3=x^3=g(x).\] 
Graph and reflect $g(x)$ to illustrate this point.

Discuss whether or not a graph could look exactly the same when we reflect across the $x$-axis. Mention that we gave a name to all graphs that have these special properties

A function that satisfies the property $f(x)=f(-x)$ then the graph of $f$ is symmetric across the $y$-axis and $f$ is called an even function. That is, inputs $x$ and $-x$ have the same outputs.
A function that satisfies the property $f(x)=-f(-x)$ then the graph of $f$ is symmetric about the origin and $f$ is called an odd function. That is, the outputs of inputs $x$ and $-x$ are the negations of one-another.

You may want to do an example of how knowing a function is even or odd can give you more information about points.

• Example:

* Suppose that $f(x)$ is even and $f(3)=6$. What is $f(-3)$? 
* Since f(x) is even, we know that f(x)=f(-x), so f(-3)=f(--3)=f(3)=6

Make it clear to your students that in general, graphing $f(-x)$ does not mean you should expect to get the same graph back, but that functions that satisfy this property are called even functions. Some students confuse the two ideas. To this end, it may be valuable to provide non-examples of even and odd functions to help students understand that being even and odd are special properties that not all functions have, and that one can determine if a function is even or odd by applying specific reflections and seeing the result.

Have students do Problems 7-10.

Students may be confused about how to find values for $f(0)$ in Problems 7 and 8. Once you feel that your students have had enough time to think about it, you can tell them that odd functions must always pass through the point $(0,0)$. To justify this, you could show them the following calculation:

Suppose $f(x)$ is an odd function. We know that $f(x)=-f(-x)$, and so $f(0)=-f(0)$. However, the only number that is the negative of itself is 0, hence $f(0)=0$. If they are still not seeing it, you may solve for it directly. Letting $y=f(0)$, we have that $y=-y$, so $2y=0$, so $y=0$ by dividing by 2. Therefore every odd function must pass through the origin.

You can then contrast this to even functions by graphing a few examples of even functions, some of which pass through the origin and some that may not. The key take away is that for an even function, we have that $f(0)$ could be any number, as the condition for being even only requires that $f(-0)=f(0)$, which is always true.