Odd or Even?

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SWBAT recognize odd and even functions from a graph or expression.

Big Idea

Just what is an odd or even function anyway? Help your students answer this question with this lesson.

Set the Stage

5 minutes

By now my students are fairly comfortable with functions and their graphs, so I begin class by asking them to graph f(x) = x^2  and f(x) = x^4 simultaneously and compare the two graphs. (MP1) After a minute or two I ask for volunteers to explain the similarities and differences of the graphs.  Usually someone observes that the graphs have almost the same shape but rarely do they notice that both graphs are symmetric about the y-axis.  I ask questions like "Are there any symmetries to either graph?" to guide them to that recognition.  I then explain that functions that are symmetric about the y-axis are called "even" functions and those that are symmetric about the origin are called "odd" functions and tell my students that we'll be working with more odd and even functions today.

Put It Into Action

40 minutes

For the first part of this section I give my students several equations to graph and look for patterns.  I only use the equations with positive exponents initially allowing several minutes for them to graph and compare them all.  After a few moments I ask them to pair-share their observations, then randomly select students to talk about what they discussed.  Most students quickly see that all their graphs take one of two forms and usually at least a few students observe that the functions with odd exponents all look the same and the ones with even exponents all look the same. (MP8) If nobody comments on the symmetries I again lead the discussion in that direction, then give them the equations with negative exponents to check out.

When everyone is comfortable with the appearance and symmetries of odd and even functions I give them another challenge.  I give each student a notecard and ask them to create two functions with at least two terms each and be ready to share them with the class. (MP2)  After a few minutes or when everyone is ready I collect the notecards and randomly select functions to post on the board.  I tell them to determine whether each function is odd, even, or neither based on its symmetry and randomly select students to answer for each function posted. (MP1)  This practice strengthens their ability to recognize odd and even functions but also lets them see that it's not always easy to tell just from looking at a graph.  I offer an alternative and show them a test for checking a function using:    odd if g(-x) = -g(x)      and        even if g(-x) = g(x)

I walk through a few examples using the functions posted on the board then give my students the Odd or Even handout and let them practice using whichever tool, graphs or algebra works best for them. (MP1, MP5)  When everyone is done or when we're down to the last ten minutes of class I have my students self-check their answers while I go through the problems.  Since I walk around while they're working, I already have a good idea of which students are struggling without ever having to collect and grade this assignment.  You could also add a connection piece about transformations here, but I think that would stretch this into at least a two day lessons for most students.


Wrap It Up

5 minutes

To wrap up this lesson I have my students write a definition of odd and even functions in their own words for their notes.  Because this is new material it will help to have a good reference in the future, so I suggest a compare-contrast of the algebraic and graphical "tools" for checking a function. (MP6)  My video discusses further why I specifically require my students to write this definition now.  An example of this might be:

"An even function is symmetric on the y-axis (think of mirror image produced by folding your paper along the y-axis ). An odd function is symmetric about the origin (you can’t really fold the paper on a point but think of reflecting each point across the x-axis and then the y-axis, so points in the 1st quadrant end up in the 3rd quadrant, etc.)."