Once you know $t$ for 2000 and 2010, treat those as the $t$-coordinates of two points. How do you compute the slope (rate of change) from two points on a line?

A linear function can be written as $P(t) = mt + b$. You already know the slope $m$. What is $P(0)$ from the table, and how does that help you find $b$?

For each answer choice, plug in $t=0$ and $t=10$. Which one gives populations of 862 and 846 for those inputs?

In one expression line, type (846-862)/(10-0) and press Enter. The value Desmos shows is the rate of change (slope) of the population in people per year between 2000 and 2010.

In a new line, type y = ((846-862)/(10-0))x + 862. Treat $x$ as $t$ (years after 2000). This line represents the linear population model through the two data points. Compare the slope and the intercept from this equation to each answer choice and select the one whose formula has the same slope and initial value.

The problem says $t$ is the number of years after 2000.

• In 2000, $t=0$ and the population is 862.
• In 2010, $t=10$ and the population is 846.

So the two points on the line are $(0,862)$ and $(10,846)$, where $t$ is the input and population is the output.

For a linear relationship, the slope is

So the population is decreasing by 1.6 people per year. Any correct function must have slope $-1.6$ with respect to $t$.

In slope-intercept form, a linear function is $P(t) = mt + b$, where $m$ is the slope and $b$ is the value when $t=0$.

• We already found $m = -1.6$.
• When $t=0$ (the year 2000), the population is 862, so $P(0) = b = 862$.

Therefore the linear model is

This matches the correct answer choice.