The bar graph shows the population of the United States, in...

The bar graph shows the population of the United States, in millions, for seven selected years. (GRAPH CANNOT COPY) Here are two functions that model the data: $M(x)=1.48 x+115.1$ $F(x)=1.44 x+120.9$ Use the functions to solve a. Write a function that models the total U.S. population for the years shown in the bar graph. b. Use the function from part (a) to find the total U.S. population in 2010 . c. How well does the result in part (b) model the actual total U.S. population in 2010 shown by the bar graph?

The bar graph shows the population of the United States, in millions, for seven selected years.
(GRAPH CANNOT COPY)
Here are two functions that model the data:
$M(x)=1.48 x+115.1$
$F(x)=1.44 x+120.9$
Use the functions to solve
a. Write a function that models the total U.S. population for the years shown in the bar graph.
b. Use the function from part (a) to find the total U.S. population in 2010 .
c. How well does the result in part (b) model the actual total U.S. population in 2010 shown by the bar graph?

Key Concepts

Evaluating Functions

Evaluating a function involves substituting a specific value for the independent variable (x) to calculate the corresponding output. This is a key step in using models for prediction, as it allows one to estimate the outcome for a particular input, such as the population in a given year.

Model Accuracy and Comparison

After using a model to generate an estimate, it is important to compare the modeled result with actual data to determine the model's accuracy. This step involves assessing how close the predicted value is to the real-world measurement, thereby evaluating the effectiveness and limitations of the model in representing the data.

Linear Functions

A linear function represents a relationship with a constant rate of change, expressed in the form f(x) = mx + b. In modeling, such functions are used to approximate data trends, where m is the slope indicating the rate of increase or decrease, and b is the y-intercept, which represents the starting value when x = 0.

Combining Functions

When multiple linear functions represent different parts or aspects of a system, they can be combined—often by addition—to create a single function that models the total effect or overall population. This process involves adding the corresponding expressions of the individual functions to generate a new linear function that reflects the aggregate behavior.

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