Question

How can a function do both of these things?

Two students in your class, Hunter and Maggie, are disputing a function. Hunter says that for the function, between `x = −2` and `x = 2`, the average rate of change is `0`. Maggie says that for the function, between `x = −2` and `x = 2`, the graph goes up through a turning point, and then back down. How can Hunter and Maggie both be correct?

Answer 1

Consider the function `f(x)=-x^2`.

Explanation:

The average rate of change is given by

`R=(f(b)-f(a))/(b-a)`

For `R=0,` we require that `f(b)-f(a)=0`, or, `f(b)=f(a)`.

Here, `b=2, a=-2,` so we want

`f(2)=f(-2)`

Quadratic functions behave in this way due to their `y-`axis symmetry; for a quadratic in the form `f(x)=ax^2, f(-x)=f(x)` .

For example, let us consider the function `f(x)=-x^2`

`f(2)=-2^2=-4`

`f(-2)=-(-2)^2=-4`

So, the average rate of change of this function is zero, as `f(2)=f(-2) -> f(2)-f(-2)=0` . Furthermore, at `x=0, f(0)=0,` so the function hits a turning point at `0` and goes back down to `-4` at `x=2:`

graph{-x^2 [-10, 10, -5, 5]}