The students in a class that meets in my room were working on a few problems at the beginning of their class period. I overhead one tell the other that an odd function goes through the origin.
I suggested to the teacher that she might try sending a True/False question to start the class discussion for the day: An odd function must go through the origin.
And so she did.
And the results were the following. (Note that the teacher deselected “Show Correct Answer” before she displayed the results for the class.)
There was a lot of talk about an odd function having a graph with symmetry about the origin. It took a long time for someone to find a counterexample to the statement. They were in the family of polynomials – and rightly so, the name of their unit was Polynomials. I kept waiting for someone to go back to their trigonometric functions.
But they didn’t. Finally someone asked about f(x)=1/x. A hyperbola. That doesn’t go through the origin. But that is an odd function. How do we know? We can show that –f(-x)=f(x), and the graph is symmetric about the origin.
What type of odd functions must go through the origin? Will every polynomial odd function go through the origin?
And a possible variation: How would the conversation have played out if the teacher had shown the correct answer and told students to determine an odd function that doesn’t go through the origin?
And so the journey continues, searching for questions that push students’ thinking and probe for misconceptions. And every once in a while, we find one that we need to share with others.
Identify the effect on the graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.