Question

Given the function f(x) = 4x + 1, EXPLAIN and show how to find the average rate of change between x = 1 and x = 3.

Answer 1

Average rate of change is another word for the slope. The function you have is presented in slope-intercept form (y=mx+b), where m=your slope. This means that you already know the slope, which is 4.

Just in case you need to show why the slope = 4, I’ll leave how to find it here:

To find the slope, use the formula: y2-y1/x2-x1

First, you need to find the y-values at x=1 and x=3. To do this, plug the x-values into the equation. The value you end up with is your y-value.

f(1) = 4(1) + 1 = 5
f(3) = 4(3) + 1 = 13

So your two points are (1, 5) and (4, 13). Now plug these points into the slope formula

m = (13-5)/(3-1) = 8/2. 8 divided by two is 4, so your average rate of change is 4. You can double check this by graphing.

Answer 2

The average rate of change is calculated as the slope of the line between the two points on the graph. Since at x = 1, y = -3, and at x = 3, y = 1, these are the two points which we calculate the slope from. Using the formula

m = (y2 - y1)/(x2 - x1)

m = (1 - (-3))/(3 - 1)

m = 4/2

m = 2

Therefore the slope, or the average rate of change, between these points is 2.