To find a linear function that models the population data, we first need to determine two points that we can use to formulate a linear equation of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
From the provided data, we can choose two points. For this example, let’s use the years 2000 and 2100:
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For the year 2000 (\( x = 0 \)):
- Population = 284.1 million → point (0, 284.1)
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For the year 2100 (\( x = 100 \)):
- Population = 577.8 million → point (100, 577.8)
Step 1: Calculate the slope \( m \)
The slope \( m \) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the points we have: \[ m = \frac{577.8 - 284.1}{100 - 0} = \frac{293.7}{100} = 2.937 \]
Step 2: Find the y-intercept \( b \)
Now we can use one of the points to calculate \( b \). We'll use the point (0, 284.1): \[ f(0) = m(0) + b \implies 284.1 = 2.937(0) + b \implies b = 284.1 \]
Step 3: Write the linear function
Now we can write the linear function that models the population: \[ f(x) = 2.937x + 284.1 \]
Conclusion
The linear function that models the population of the country based on the data provided is: \[ f(x) = 2.937x + 284.1 \] where \( x \) is the number of years after 2000, and \( f(x) \) is the population in millions.