The rate of change of a function is the ratio between the changes in the y-value and the changes in the x-value.
So, let's evaluate each function.
(a)
Let's find the point in which x = 0 and x = 2:
(0, 1)
(2, 4)
Then, the rate (r) of change is:
r = (4-1)/(2-0)
r = 3/2
(b)
Let's find the point in which x = 0 and x = 2:
(0, 0)
(2, 2)
Then, the rate of change is:
r = (2 - 0)/(2 - 0)
r = 2/2
r = 1
(c)
Let's find the point in which x = 0 and x = 2:
(0, -1)
(2, 0)
Then, the rate of change is:
r = (0 - (-1))/(2 - 0)
r = 1/2
r = 1/2
(d)
Let's find the point in which x = 0 and x = 2:
(0, 0.5)
(2, 2.5)
Then, the rate of change is:
r = (2.5 - 0.5)/(2 - 0)
r = 2/2
r = 1
The function that has the greatest rate of change over the interval [0, 2] is (a).