Question

Interval does h have a negative average of change

Answer 1

The average rate of change over some interval [a, b] is equal to the slope of the secant line from (a, h(a)) to (b, h(b)).

h(t) is a quadratic function, so its graph is a parabola, and in particular it's one that has a minimum of -4 when t = 2.

The secant line over an interval [a, b] will have a negative slope if the distance from a to 2 is larger than the distance from b to 2.

(A) If a = 4 and b = 5, then |a - 2| = 2 and |b - 2| = 3, so the slope and hence average rate of change is positive.

(B) If a = -1 and b = 5, then |a - 2| = 3 and |b - 2| = 3, so this ARoC is zero.

(C) If a = 0 and b = 4, then |a - 2| = 2 and |b - 2| = 2, so this ARoC is also zero.

(D) If a = -1 and b = 4, then |a - 2| = 3 and |b - 2| = 2, so this ARoC is negative.