Question

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

Tiles:
f(x) = x2 + 3x
interval: [-2, 3]

f(x) = 3x - 8
interval: [4, 5]

f(x) = x2 - 2x
interval: [-3, 4]

f(x) = x2 - 5
interval: [-1, 1]

Answer 1

By definition, the average change of rate is given by:

`AVR = (f(x2)-f(x1))/(x2-x1)`
We will calculate AVR for each of the functions.
We have then:

f(x) = x^2 + 3x interval: [-2, 3]:

`f(-2) = x^2 + 3x = (-2)^2 + 3(-2) = 4 - 6 = -2 f(3) = x^2 + 3x = (3)^2 + 3(3) = 9 + 9 = 18`

`AVR = (-2-18)/(-2-3)`

`AVR = (-20)/(-5)`

`AVR = 4`

f(x) = 3x - 8 interval: [4, 5]:

`f(4) = 3(4) - 8 = 12 - 8 = 4 f(5) = 3(5) - 8 = 15 - 8 = 7`

`AVR = (7-4)/(5-4)`

`AVR = (3)/(1)`

`AVR = 3`

f(x) = x^2 - 2x interval: [-3, 4]

`f(-3) = (-3)^2 - 2(-3) = 9 + 6 = 15 f(4) = (4)^2 - 2(4) = 16 - 8 = 8`

`AVR = (8-15)/(4+3)`

`AVR = (-7)/(7)`

`AVR = -1`

f(x) = x^2 - 5 interval: [-1, 1]

`f(-1) = (-1)^2 - 5 = 1 - 5 = -4 f(1) = (1)^2 - 5 = 1 - 5 = -4`

`AVR = (-4+4)/(1+1)`

`AVR = (0)/(2)`

`AVR = 0`


Answer:
these functions from the greatest to the least value based on the average rate of change are:
f(x) = x^2 + 3x
f(x) = 3x - 8
f(x) = x^2 - 5
f(x) = x^2 - 2x