To find a linear function that models the population data, we need to determine the slope $m$ and the y-intercept $b$ of the line in the form:

$f(x) = mx + b$

where:

• $x$ is the number of years after 2000,
• $f(x)$ is the population in millions.

We can use two points from the table to calculate the slope $m$. Let's take the year 2000 ($x = 0$, $f(x) = 282.9$) and the year 2100 ($x = 100$, $f(x) = 574.8$).

Step 1: Calculate the slope $m$

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Using the points $(0, 282.9)$ and $(100, 574.8)$:

$m = \frac{574.8 - 282.9}{100 - 0} = \frac{291.9}{100} = 2.919$

Step 2: Find the y-intercept $b$

The y-intercept is the population at $x = 0$, which is already given as $f(0) = 282.9$. So:

$b = 282.9$

Step 3: Write the linear equation

The linear function is:

$f(x) = 2.919x + 282.9$

Final Answer:

The linear function is:

$f(x) = 2.919x + 282.9$

Would you like further details on how the linear model works or have any questions?

Here are 5 related questions you might find useful:

1. How would the equation change if we used different years to calculate the slope?
2. How do we interpret the slope of 2.919 in this context?
3. What does the y-intercept represent in terms of population data?
4. Can this model accurately predict populations for years outside the given range?
5. How could we improve this model if the population growth is not perfectly linear?

Tip: When modeling with linear functions, ensure the data fits the assumption of constant growth over time to get accurate results.