Question

Please help! :)
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

Consider this graph...

the rate of change between x=-1 and x=1 is 0 because they have the same y-value.
the graph goes up to a turning point (at x=0) and then comes back down.

Answer 2

So the average rate of change is 0 because the y-value is the same?

Answer 3

yes...rate of change is another name for a slope of a line. if you've studied lines and linear equations, you may remember that the slope is found to be
\[m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\]

so if the y-values are the same for any two points, then the rate of change between those points will be 0.

make sense?

Answer 4

yes! so in other words Wilson and Alexis can both be correct because the y-value is the same?

Answer 5

almost...that's really only half of it.

Wilson can be correct if, for the function, f(-1) = f(1)...meaning the y-values are the same at x=-1 and x=1

Alexis can be corret if the graph, between x = -1 and x = 1 does what she claims.

Do you know what kind of function has a graph like the one i drew? it would be useful to include that in your explanation...

Answer 6

Quadratic?

Answer 7

excellent! :) So if the function is quadratic opening down (a requirement for Alexis' claim) with the vertex occurring at x=0 (a requirement for Wilson's claim) then f(-1) = f(1) meaning the average rate of change is 0 between x=-1 and x=1. Since it opens down, it increases between x=-1 and x=0, turns around at x=0, and decreases from x=0 to x=1.

:)

Answer 8

Thank you so much!

Answer 9

you are quite welcome! :)