families of functions

figure 1: the family of functions with b=2 and a=1 (green curve) and 3 (blue curve)

figure 2: the family of functions with a=3 and b=1 (green curve) and 3 (blue curve)

figure 3: the family of functions with a=-1 (green curve), and -2 (blue curve) and b=2.

An example of how we might look at families of functions. Let's consider the family given by

We know that this is a family of functions because of the parameters a and b (which are unspecified constants: we know they're constant, but don't know what they are. And suppose that we're asked to find what functions in the family have a local minimum at (1,-1).

A good place to start with this type of question is to graph the function for some values of the parameters. This is done in the figures to the right. What do you notice? Are we going to get a local minimum at (1,-1) with these parameter choices?

Nope! We obviously are going to need a to be negative! Good to know. (Why is this? It's because we know by multiplying a function by a negative value, which is what taking a to be negative does, reflects it across the x-axis.)

Ok, how do we find the location of the local minimum to make sure that it's at (1,-1)? The local minimum will be at a critical point, which is where y'(x) = 0. So let's find the derivative and see what that says.

So if we set this to zero, we get

We don't want a = 0, and the exponential can never equal zero, so this says that x = 1/b. Therefore if we want the minimum to be at x = 1 we must have b = 1!

Finally, we want the value at the minimum to be y = -1. So that says

So a = -e !

This says that there is in fact only one member of the family that has a local minimum at (1,-1), and that member is the function

Check that this works by graphing it on your calculator!