Question

1. given the graph to the right, find the average rate of change between:

a. points a & b
b. points b & c
c. x = - 2 and x = 0

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Assume coordinates from the graph (since no values given, general form)

Let the coordinates of point $A$ be $(x_A,y_A)$, point $B$ be $(x_B,y_B)$ and point $C$ be $(x_C,y_C)$.

a. For points $A$ and $B$

The average rate of change between $A$ and $B$ is $\frac{y_B - y_A}{x_B - x_A}$.

b. For points $B$ and $C$

The average rate of change between $B$ and $C$ is $\frac{y_C - y_B}{x_C - x_B}$.

c. For $x=-2$ and $x = 0$

Let $y_1$ be the value of the function at $x=-2$ and $y_2$ be the value of the function at $x = 0$. The average rate of change is $\frac{y_2 - y_1}{0-(-2)}=\frac{y_2 - y_1}{2}$.

Answer:

a. $\frac{y_B - y_A}{x_B - x_A}$
b. $\frac{y_C - y_B}{x_C - x_B}$
c. $\frac{y_2 - y_1}{2}$ (where $y_1$ is the function - value at $x=-2$ and $y_2$ is the function - value at $x = 0$)