Question

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

1.
f(x) = x2 + 3x
interval: [-2, 3]
2.
f(x) = 3x - 8
interval: [4, 5]
3.
f(x) = x2 - 2x
interval: [-3, 4]
4.
f(x) = x2 - 5
interval: [-1, 1]

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