Question

What is the average rate of change of f(x) = −2x + 1 from x = −5 to x = 1?

Answer 1

The rate of change is the slope. m = slope. We can see that the rate of change is going to be -2 because the function f(x) = −2x + 1 is in the slope intercept form and is linear, which slope intercept form is y = mx + b. Remember m = slope = rate of change.

There are a few methods to finding the average rate of change. A popular one is using the slope formula:

Slope Formula
m = slope = rate of change


`m = (Y_2 - Y_1)/(X_2 - X_1)`



Now we know the formula for rate of change, we can use this but first to use this formula we need some points. We get our points by plugging in -5 and 1 into our equation f(x) = -2x + 1

f(x) = -2x + 1
f(x) = -2(-5) + 1
f(x) = 10 + 1
f(x) = 11
So y = 11 when x = -5. We have our first point, which is (-5, 11)

Lets find our next point
f(x) = -2x + 1
f(x) = -2(1)+ 1
f(x) = -2+ 1
f(x) = -1

So y = -1 when x = 1. We have our second point, which is (1, -1)

Now we can find the rate of change by using our slope formula and our two points: (-5, 11) and (1,-1)


`m = (Y_2 - Y_1)/(X_2 - X_1)`

`m = (11 - (-1))/((-5) - 1)`

`m = (12)/(-6)`

`m = -2`

rate of change = -2