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
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Explanation:

Step1: Determine the linear function form

We know that the function is linear (since we found the rate of change, which is the slope, \(m = 20\)). The general form of a linear function is \(y=mx + b\), where \(m\) is the slope (rate of change) and \(b\) is the initial value (y - intercept).

Step2: Use a point to find \(b\)

We can use one of the given points, say \((3,120)\), and the slope \(m = 20\) in the equation \(y=mx + b\).
Substitute \(x = 3\), \(y = 120\), and \(m = 20\) into the equation:
\[120=20\times3 + b\]
\[120 = 60 + b\]

Step3: Solve for \(b\)

Subtract 60 from both sides of the equation:
\[b=120 - 60\]
\[b = 60\]

Answer:

The initial value is \(60\). It represents the number of kanji characters Wesley knew at the start of summer vacation (when \(x = 0\), i.e., at week 0).