consider the functions graphed below. determine which function has the greatest rate of change over the…

Explanation:

Step1: Recall rate - of - change formula

The average rate of change of a function $y = f(x)$ over the interval $[x_1,x_2]$ is $\frac{f(x_2)-f(x_1)}{x_2 - x_1}$. Here, $x_1 = 0$ and $x_2=2$.

Step2: Analyze function $a$

From the graph, when $x = 0$, $y_a(0)=1$, when $x = 2$, $y_a(2)=4$. The rate of change of $a$ is $\frac{y_a(2)-y_a(0)}{2 - 0}=\frac{4 - 1}{2}=\frac{3}{2}$.

Step3: Analyze function $c$

When $x = 0$, $y_c(0)=1$, when $x = 2$, $y_c(2)=3$. The rate of change of $c$ is $\frac{y_c(2)-y_c(0)}{2 - 0}=\frac{3 - 1}{2}=1$.

Step4: Analyze function $d$

When $x = 0$, $y_d(0)= - 1$, when $x = 2$, $y_d(2)=1$. The rate of change of $d$ is $\frac{y_d(2)-y_d(0)}{2 - 0}=\frac{1-(-1)}{2}=1$.

Step5: Analyze function $b$

When $x = 0$, $y_b(0)=0$, when $x = 2$, $y_b(2)=4$. The rate of change of $b$ is $\frac{y_b(2)-y_b(0)}{2 - 0}=\frac{4 - 0}{2}=2$.

Answer:

D. $b$