Question

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. Explain how Hunter and Maggie can both be correct, using complete sentences.

Answer 1

From the given problem

The average rate of change of a function over an interval [a, b] is:

{ f(b) - f(a) } / (b - a) .........(1)

eg function: f(x) = ( x -2 )( x +2 )( x +5 )

If Hunter says that the average rate between [-2 2], he was referring to f(2) =(-2) and he is correct i.e the numerator in equation (1) is zero. If its continuous function has the same value of zero at two points,then the curve must turn in between.

Hence both be correct

Note: Given that the function has the same value at x=-2 and x = +2 ( in order for the average rate of change to be zero, the function must have the same value at x = -2 and x= 2). so at x = -2, the function is either increasing ( curve is growing up) or decreasing(the curve is going down)