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.
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.
