Question

The function f ( x ) = 2 x ^2 − x − 4 models the shape of a ditch. What is the average slope over the interval −4 ≤ x ≤ 2?

Answer 1

`Rate = -5`

Explanation:

Given

`f(x) = 2x^2 - x - 4`

`-4 \le x \le 2`

Required

Determine the average slope over the given interval

This is calculated as:

`Rate= (f(b) - f(a))/(b - a)`

Where:

`a = -4`

`b =2`

So:

`Rate= (f(2) - f(-4))/(2 - (-4))`

`Rate= (f(2) - f(-4))/(2 +4)`

`Rate= (f(2) - f(-4))/(6)`

Solving for f(2) and f(-4), we have:

`f(x) = 2x^2 - x - 4`

`f(2) = 2 *2^2 -2 - 4`

`f(2) = 2`

`f(-4) = 2*(-4)^2 - (-4) - 4`

`f(-4) = 2*16 +4 - 4`

`f(-4) = 32`

So, the equation becomes

`Rate= (f(2) - f(-4))/(6)`

`Rate= (2 - 32)/(6)`

`Rate= (-30)/(6)`

`Rate = -5`