Question

at the start of summer vacation, wesley decides to learn a number of kanji characters each week. 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). 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. find and describe the initial value of the function. the initial value is 60. wesley knows 60 characters at the start of summer vacation.

Explanation:

Step1: Calculate rate of change

The rate of change (slope $m$) of a linear - function passing through two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by the formula $m=\frac{y_2 - y_1}{x_2 - x_1}$. Here, $(x_1,y_1)=(3,120)$ and $(x_2,y_2)=(5,160)$. So, $m=\frac{160 - 120}{5 - 3}=\frac{40}{2}=20$. It represents the number of characters Wesley learns each week.

Step2: Find the initial - value

The equation of a line in slope - intercept form is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept (initial value). We know $m = 20$, and using the point $(3,120)$: $120=20\times3 + b$. Solving for $b$:
\[

$$\begin{align*} 120&=60 + b\\ b&=120 - 60\\ b&=60 \end{align*}$$

\]
The initial value is 60, which means Wesley knows 60 characters at the start of summer vacation.

Answer:

The rate of change is 20. It represents the number of characters Wesley learns each week.
The initial value is 60. Wesley knows 60 characters at the start of summer vacation.