Question

Arrange these functions from the greatest to least value based on the average rate of change in the specified interval.

Answer 1

Let us create a table that evaluates the slope of each function within its specified interval.

Define
 [x1,x2] = the specified interval
f1 = f(x1)
f2 = f(x2)

The average rate of change is the slope, calculated as
Slope = (f2 - f1)/(x2 - x1)


       f(x)   Interval  x2-x1           f1          f2    Slope
----------- -----------   --------   --------   ----------   ---------
x² + 3x      [-2, 3]          5         -2          18           4
3x - 8         [4, 5]          1           4            7           3
x² - 2x      [-3, 4]          7          15           8          -1
x² - 5          [-1, 1]         2          -4          -4           0

Answer:
In order of the greatest slope to the lowest slope, the order of the functions is 1, 2, 4 and 3. That is,
1. f(x) = x² + 3x
2. f(x) = 3x - 8
4. f(x) = x² - 5
3. f(x) = x² - 2x