Question

f(x) = 3 cos(2x − π) − 1 g(x) trig graph with points at 0, 0 and pi over 2, 3 and pi, 0 and 3 pi over 2, negative 3 h(x) = sin x − 4 Which function has the smallest rate of change from x = 0 to x = pi over 2?

Answer 1

The rate of change is the ratio of the change in f(x) (or g(x)) to the change in x. Here, you're given a specific interval for x, so the change in x is the same for both functions. Hence you only need to look at the change in f(x) (or g(x)).

The change in f(x) over the interval is
  f(π/2) - f(0) = (3cos(2π/2 -π) -1) - (3cos(0 -π) -1)
    = (3-1) - (-3-1)
    = 2 - (-4) = 6

The change in g(x) over the interval is
  g(π/2) - g(0) = 3 - 0 = 3

The amount of change over the interval [0, π/2] is smaller for g(x) than for f(x). So, the function with the smallest rate of change from x=0 to π/2 is g(x).