A quadratic function is given by

MATHEMATICS · Multi-part · AP · AP Precalculus · Polynomial and Rational Functions · Polynomial and Rational Functions

Problem

p(x)=x^{2}-4kx

, where

k

is a real parameter. Consider the closed interval

[k,k+2]

. (1) Express the average rate of change of

p

[k,k+2]

as a function of

k

. Simplify completely. (2) Find all real values of

k

for which the average rate of change on

[k,k+2]

equals

8

. (3) For each value of

k

obtained in part (b), determine the sign (positive, negative, or zero) of the average rate of change if

k

is perturbed slightly upward. Explain your reasoning without recomputing the entire expression.