To find a linear function that models the population data, we need to determine a linear equation of the form:
\[ f(x) = mx + b \]
where \( x \) is the number of years after 2000, \( f(x) \) is the population in millions, \( m \) is the slope of the line, and \( b \) is the y-intercept.
Step 1: Define the Points
We will use two points to calculate the slope \( m \). Let’s take the years 2000 and 2100:
- For 2000 (which corresponds to \( x = 0 \)): Population = 284.1 million
- For 2100 (which corresponds to \( x = 100 \)): Population = 577.8 million
The points are:
- \( (0, 284.1) \)
- \( (100, 577.8) \)
Step 2: Calculate the Slope \( m \)
The formula for calculating the slope \( m \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using our points:
\[ m = \frac{577.8 - 284.1}{100 - 0} = \frac{293.7}{100} = 2.937 \]
Step 3: Calculate the Y-Intercept \( b \)
Using the point \( (0, 284.1) \) to find \( b \):
\[ f(0) = 2.937 \cdot 0 + b = 284.1 \] \[ b = 284.1 \]
Final Function
Putting it all together, the linear function that models the population is:
\[ f(x) = 2.937x + 284.1 \]
Final Answer
Therefore, the function is:
- Slope: \( 2.937 \)
- Y-intercept: \( 284.1 \)
So, the final answer is:
\[ f(x) = 2.937x + 284.1 \]
(rounded to three decimal places as needed).