Question

Find the average rate of change of the function passing through (2,-1) and (-8,11)​

Answer 1

`(-6 / 5)` between the two points, given that this function is continuous.

Explanation:

Assume that the function in this question is continuous over the interval between the two given points. Divide the change in the value of this function by the width of the interval to find the average rate of change of this function over that interval.

For example, consider a continuous function that goes through
`(x_{\text{a}},\, y_{\text{a}})` and
`(x_{\text{b}},\, y_{\text{b}})`. The width of the interval between these two points is
`(x_{\text{b}} - x_{\text{a}})`. Over this interval, the value of this function has changed from
`y_{\text{a}}` to
`y_{\text{b}}`, such that the change in the value of the function is
`(y_{\text{b}} - y_{\text{a}})`.

The average rate of change of this function between these two points would be
`(y_{\text{b}} - y_{\text{a}}) / (x_{\text{b}} - x_{\text{a}})`.

The function in this question goes through the two points
`(2,\, -1)` and
`(-8,\, 11)`. Substitute in these values to find the average rate of change of this function between these two points:

`\begin{aligned} (\text{avg. rate of change}) &= \frac{y_{\text{b}} - y_{\text{a}}}{x_{\text{b}} - x_{\text{a}}} \\ &= (11 - (-1))/((-8) - 2) \\ &= (12)/(-10) \\ &= -(6)/(5)\end{aligned}`.