To create a linear function that represents the population $P$ of senior high school students over time $t$, where $t$ is the number of years after 2015, we start by determining the slope (rate of change) and the y-intercept of the line.

Step 1: Define the Variables

• Let $P(t)$ represent the population of senior high school students in year $t$.
• Let $t = 0$ correspond to the year 2015.

Step 2: Identify the Given Points

We have two data points based on the years and student populations:

1. In 2015 (when $t = 0$), the population $P = 3500$.
2. In 2018 (when $t = 3$), the population $P = 4100$.

These points can be represented as:

• $(t_1, P_1) = (0, 3500)$
• $(t_2, P_2) = (3, 4100)$

Step 3: Calculate the Slope

The slope $m$ of a linear function is given by the change in the population divided by the change in time:

$m = \frac{P_2 - P_1}{t_2 - t_1}$

Substitute the given values:

$m = \frac{4100 - 3500}{3 - 0} = \frac{600}{3} = 200$

So, the slope $m = 200$. This means the population increases by 200 students per year.

Step 4: Write the Linear Function

A linear function can be expressed in the form:

$P(t) = mt + b$

where $m$ is the slope and $b$ is the y-intercept.

Since $t = 0$ corresponds to the year 2015, and at this time $P = 3500$, the y-intercept $b$ is 3500.

Thus, the function rule that describes the population $P(t)$ as a linear function of time $t$ is:

$P(t) = 200t + 3500$

Conclusion

The linear function $P(t) = 200t + 3500$ models the population of senior high school students, where $t$ is the number of years after 2015.

Would you like more details, or do you have any questions?

Here are some related questions:

1. What will the population be in 2020 according to this model?
2. How many years after 2015 will the population reach 5000 students?
3. What does the slope of 200 represent in this context?
4. Can this model be used indefinitely? Why or why not?
5. How would the function change if the population in 2018 was different?

Tip: When interpreting linear functions, the slope represents the rate of change, and the y-intercept represents the starting value.