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.the rate of change is 20 , it represents the number of characters wesley learns each week .2) find and describe the initial value of the function.the initial value is 60 ,wesley knows 60 characters at the start of summer vacation.■ write an equation to represent the total number of characters wesley knows, $y$.

Explanation:

Step1: Calculate rate of change

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

Step2: Find initial value (y-intercept)

Use slope-intercept form $y=mx+b$, substitute $m=20$, $x=3$, $y=120$:
$$120=20\times3+b$$
$$120=60+b$$
$$b=120-60=60$$

Step3: Write the function

Substitute $m=20$ and $b=60$ into $y=mx+b$:
$$y=20x+60$$

Answer:

• Rate of change: $20$, represents the number of characters Wesley learns each week.
• Initial value: $60$, represents the number of characters Wesley knows at the start of summer vacation.
• Function: $y=20x+60$