Question

the total number of characters wesley knows, $y$, is a function of the number of weeks, $x$. the graph of the function passes through the points $(3, 120)$ and $(5, 160)$.(1) find and describe the rate of change of the function.(2) the rate of change is 20. it represents the number of characters wesley learns each week.(3) find and describe the initial value of the function.(4) the initial value is 60.(5) wesley knows 60 characters at the ? of summer vacation.startend

Explanation:

Step1: Calculate rate of change

The rate of change (slope) $m$ between points $(x_1,y_1)=(3,120)$ and $(x_2,y_2)=(5,160)$ is calculated as:
$$m=\frac{y_2-y_1}{x_2-x_1}=\frac{160-120}{5-3}=\frac{40}{2}=20$$

Step2: Find the linear function

Use point-slope form $y-y_1=m(x-x_1)$ with $(3,120)$:
$$y-120=20(x-3)$$
Simplify to slope-intercept form $y=mx+b$:
$$y=20x-60+120=20x+60$$

Step3: Identify initial value context

The initial value is the $y$-intercept $b=60$, which occurs when $x=0$ (start of the vacation, 0 weeks passed).

Answer:

1. Rate of change: 20, representing the number of characters Wesley learns each week.
2. Function: $y=20x+60$
3. Initial value: 60, which is the number of characters Wesley knows at the start of summer vacation.