Question

Two students in your class, Wilson and Alexis, are disputing a function. Wilson says that for the function, between x = -1 and x = 1, the average rate of change is 0. Alexis says that for the function, between x = -1 and x = 1, the graph goes up through a turning point, and then back down. Explain how Wilson and Alexis can both be correct, using complete sentences.

Answer 1

We are given that the Wilson states that for the function f(x), -1≤x≤1 the average rate of change of the function is 0 i.e.
`(dy)/(dx) =0`.

Now, when we have
`(dy)/(dx) =0`, this implies that a function assume a critical point -1≤x≤1.

If a function has a critical point, we get that at that point the function either assumes its maximum or minimum value.

On the other hand, Alexis states that for the function f(x), -1≤x≤1 the graph goes up through a turning point and then goes back down.

This gives us that the function has a maximum value in -1≤x≤1 from where the graph changes its position from upward to downward.

So, we see that both Wilson and Alexis are implying that the function f(x) has a maximum value in -1≤x≤1.

Hence, both of them are correct in their statements.

Answer 2

Well if we analyze the formula for average rate of change (y1-y2/x1-x2) then we know the change I'm y values must be zero because we already know that the change in x values is 2. This means that if there is a turning point between x=-1 and x=1 then as long as the y values at these points are the same, then the average ROC is zero. For example, if the y values are 2, 2-2 is zero, zero divided by 2 (change In x) is still zero