Question

Consider the function: g(x)=−6x

2
−8x−3 Find the average rate of change of g(x) between the points x=−1 and x=4. Give your answer as an integer or reduced fraction.

Answer 1

The average rate of change is -26.

Explanation:

This question has already been correctly answered, but I'll add a graphic (see attached graph) with explanation.

First, let's format the equation correctly: g(x)=−6x^2−8x−3

We want the average rate of change, or slope, of the parabola between points x = -1 and x = 4. First, calculate the corresponding y values for both points by entering the x values into the equation. The two points are:

(-1, 1) and (4, -133)

We'll connect those two points.

The line between these two points has a slope that is the average between the two points on the curve.

We'll look for a line with the form y-mx+b, where m is the slope and b the y-intercept.

The slope is the Rise/Run of the line.

Rise = (-131 - (-1)) = -130

Run = (4 - (-1)) = 5

Slope = Rise/Run - (130/5)

Slope = -26

The average rate of change for the interval of x from -1 to 4 is -26.

Out of curiosity, lets also find the y-intercept, b.

Use the slope in the equation y=mx+b:

y = -26x+b

Now enter one of the two points and solve for b. I used (-1, 1)

y = -26x+b

1 = -26(-1)+b

1 = 26 + b

b = -25

The line is -26x-25

Answer 2

- 26

Explanation:

the average rate of change of g(x) in the closed interval [ a, b ] is

`(g(b)-g(a))/(b-a)`

here interval is [ - 1, 4 ] , then

g(b) = g(4) = - 6(4)² - 8(4) - 3 = - 6(16) - 32 - 3 = - 96 - 35 = - 131

g(a) = g(- 1) = - 6(- 1)² - 8(- 1) - 3 = - 6(1) + 8 - 3 = - 6 + 5 = - 1

Then

average rate of change =
`(-131-(-1))/(4-(-1))` =
`(-131+1)/(4+1)` =
`(-130)/(5)` = - 26