Calculate the average rate of change of the function $f(x) = 2x + 3$ between the points $(1, f(1))$ and $(3, f(3))$.
Answer
Finally, we calculate the average rate of change, which is the difference in the y-coordinates divided by the difference in the x-coordinates: $\frac{f(3) - f(1)}{3 - 1} = \frac{4}{2} = 2$.
Step 1 ：First, we need to calculate the y-coordinate for each point. We substitute $x$ into $f(x)$ to get $f(1) = 2(1) + 3 = 5$ and $f(3) = 2(3) + 3 = 9$.
Step 2 ：Then, we find the difference in the y-coordinates of these two points, which is $f(3) - f(1) = 9 - 5 = 4$.
Step 3 ：Next, we find the difference in the x-coordinates of these two points, which is $3 - 1 = 2$.
Step 4 ：Finally, we calculate the average rate of change, which is the difference in the y-coordinates divided by the difference in the x-coordinates: $\frac{f(3) - f(1)}{3 - 1} = \frac{4}{2} = 2$.
