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 $(5, 160)$.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 $square$.

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 $\frac{y_2-y_1}{x_2-x_1}$.
$$m=\frac{160-120}{5-3}=\frac{40}{2}=20$$

Step2: Write point-slope form

Use point $(3,120)$ to set up the equation: $y-y_1=m(x-x_1)$
$$y-120=20(x-3)$$

Step3: Simplify to slope-intercept form

Expand and solve for $y$ to find the function.
$$y=20x-60+120=20x+60$$

Step4: Identify initial value

The initial value is the $y$-intercept (when $x=0$).
$$y=20(0)+60=60$$

Answer:

Rate of change:

The rate of change is $20$. It represents the number of characters Wesley learns each week.

Function:

$y=20x+60$

Initial value:

The initial value is $60$