Question

which of the following graphs has the largest average rate of change from -1 to 2?

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ from $x=a$ to $x = b$ is given by $\frac{f(b)-f(a)}{b - a}$. Here, $a=-1$ and $b = 2$, so the formula is $\frac{f(2)-f(-1)}{2-(-1)}=\frac{f(2)-f(-1)}{3}$.

Step2: Estimate function values from graphs

For each graph, estimate the values of $f(2)$ and $f(-1)$ and calculate $\frac{f(2)-f(-1)}{3}$.

• For the first graph: Let's assume $f(-1)\approx 1$ and $f(2)\approx 1$, then $\frac{f(2)-f(-1)}{3}=\frac{1 - 1}{3}=0$.
• For the second graph: Assume $f(-1)\approx - 2$ and $f(2)\approx 4$, then $\frac{f(2)-f(-1)}{3}=\frac{4-(-2)}{3}=\frac{6}{3}=2$.
• For the third graph: Assume $f(-1)\approx 4$ and $f(2)\approx 4$, then $\frac{f(2)-f(-1)}{3}=\frac{4 - 4}{3}=0$.
• For the fourth graph: Assume $f(-1)\approx 1$ and $f(2)\approx 1$, then $\frac{f(2)-f(-1)}{3}=\frac{1 - 1}{3}=0$.

Answer:

The second graph has the largest average rate of change from - 1 to 2.