Experience First:

Symmetry is a topic that has been of interest to even the earliest civilizations with important applications in the fields of science and nature, art, music, literature, architecture, and of course, mathematics. Today we dive into the meaning of even and odd functions by noticing patterns in graphs, making conjectures, and forming generalizations. Students are asked to notice features in the graph and connect this to ordered pairs in a table. They are then asked to devise their own method for checking algebraically if a given function is even or odd. Connecting multiple representations is a key aspect of this lesson! The last question is meant to be fun and can be answered in a variety of ways. Ask students to justify why Anne Hathaway’s face is even or odd. Encourage students to use evidence from the ordered pairs in the photo. Because no face is perfectly symmetrical, students may differ in their judgment of if her face is “even” or not, based on the strict mathematical definition.

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Monitoring Questions:

• Do all quadratics have y-axis symmetry?

• How many points would you need to be convinced that a function has y-axis symmetry (if you didn’t see its graph)?

• Is it possible for a graph to pass through (-4, 8) and (4,-8) and not have origin symmetry?

• What do you think we call a function with x-axis symmetry?

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Formalize Later:

This lesson provides a great opportunity to tie in ideas from Geometry about transformation. Most Geometry courses look at ordered pair rules for various transformations, such as (x,y) →(-x,y) for reflection across the y-axis. Ask students to consider why origin symmetry is the same as 180˚ rotational symmetry, tying back to the ordered pair rule of (x,y) → (-x,-y). A further connection is that origin symmetry represents a function that has been reflected over the y-axis AND the x-axis (again, tie back to the ordered pair rules). 

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An interesting question to ask students is why there’s not a word for a function with x-axis symmetry. Students should arrive at the conclusion that such a “function” does not exist, since a relation with x-axis symmetry is by definition, not a function.

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Some misconceptions with even and odd functions appear when students generalize that all functions with an even degree are even and all functions with an odd degree are odd. We parse out some of these claims in the Check Your Understanding questions.

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Decide if you will require students to give an algebraic proof for why a function is even and odd or simply provide evidence of a function being even or odd. The latter can be done with some ordered pair examples, but the former requires algebraic manipulation of expressions when evaluating functions at -x.

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