PreCalculus Symmetry (odd, even or neither)

Our final function characteristic is symmetry. This is largely visual at first.










We start off by identifying letters of the alphabet that have vertical line symmetry, horizontal line symmetry and multiple line symmetry. I write those headings on my whiteboard and have students go up and draw examples with the dotted lines of symmetry. (oops forgot to take a picture of this!).


Then we talk about how we can do the same with functions but can't just do any horizontal or vertical line when discussing symmetry. We need a standard horizontal line to be symmetrical to & a standard vertical line to be symmetry. Aha - how about those axes? So then students draw some visuals for both of those - soon realizing that symmetry with the x-axis results in something that is not a function. For multiple line symmetry - we talk about symmetry with respect to the origin (or a double flip to get back to the original - a flip over each axes - helps them visualize it). So we figure out what parent functions appear to be symmetrical and what type they have.

We use a foldable to organize our information in this lesson.


Next is getting the vocab down - we have EVEN and ODD functions. Interesting names, no? Wonder what relevance they have.....


Then nasty little teacher as I am I point out - oh no, remember that we can't take what we "see" at face-value. A visual is not enough to prove anything mathematically. I tell students that we have to verify in a general way algebraically, following a set format. Sort of mini-proof like. Some students get pretty annoyed at this, why do we have to do that? Hey, all I can say is "justify, justify, justify". If I'm not convinced then it's not true.

Of course there is always the student who doesn't believe me when I tell them they must have each part. Sigh. They do. And if they don't it will be reflected in their grade. Sigh. 

My students learn that I am a big fan of shortcuts but I also insist on precision. 

All precalculus foldables are found here.