Monday, a student flagged me down for some help with exponential functions. “They give me two points that the function passes through, and they want me to give the equation of the function.” Using my typical Socratic TutorFu I asked, “Okay, what do you want to try?” “Well, we’ve got two points, so let’s try to find the slope between them.” I’m a little stumped. Is this a new technique I don’t know? “Umm, okay. Can you explain why?” She looks at me blankly. “Because that’s what we always did for the equations for lines…?”
In related news, I was musing about exponential functions. The generic form is:
First, what’s up with Q and t? x and y are old school?
Then I found myself thinking about the formula for compound interest, at which point I wondered why the generic form of the exponential doesn’t include a coefficient for the exponent, like so:
Thankfully, I thought about it for a minute rather than just asking another tutor. Oh, right. That can be modified to:…
… at which point bk can be reduced to a single term. Not quite a “duh” moment, but I’m glad I figured it out myself.
But here’s a topic I’m not doing so well with. Fairly early in algebra, around the time kids are learning to graph lines, math texts start talking about functions. They expect students to know what a function is, and to know about the vertical line test. They classify variables as “input variables” and “output variables”, and that one input can’t have more than one output (which is just another way of stating the vertical line test).
I can’t help but wonder, who cares?!
Not in the “Who cares what X is?!” sense, but why is this topic so important? Why is this distinction so critical at this stage of algebra education?
Here are two equations:
Both can be manipulated algebraically in the exact same fashion. You can solve either of them for y and for x. You can graph both of them. But one is a function and the other isn’t. And I fail to see why that’s critically important. Am I missing something big here?