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

The bar graph shows the population of the United States, in millions, for six selected years. (Graph cannot copy) Here are two functions that model the data: $$ \begin{array}{l}{M(x)=1.54 x+114.6} \\ {F(x)=1.48 x+120.6}\end{array} $$ Use the functions in the previous column 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 2005. c. Does the result in part (b) overestimate or underestimate the actual total U.S. population in 2005 shown by the bar graph? By how much?

The bar graph shows the population of the United States, in millions, for six selected years.
(Graph cannot copy)
Here are two functions that model the data:
$$
\begin{array}{l}{M(x)=1.54 x+114.6} \\ {F(x)=1.48 x+120.6}\end{array}
$$
Use the functions in the previous column 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 2005.
c. Does the result in part (b) overestimate or underestimate the actual total U.S. population in 2005 shown by the bar graph? By how much?

Key Concepts

Linear Functions

Linear functions are mathematical models represented by equations of the form f(x) = mx + b, where m is the slope and b is the y-intercept. They model relationships where the rate of change between variables is constant, making them ideal tools for representing phenomena like steady population growth over time.

Function Combination

Function combination involves adding or otherwise combining separate functions to create a new model that represents a composite quantity or relationship. In this context, summing individual population models to form a total population function is an application of function addition that allows for analysis of combined data.

Function Evaluation

Function evaluation is the process of substituting a specific value for the variable in a function to calculate the corresponding outcome. This concept is crucial for predicting values at specific time points, such as calculating the population in a particular year using a mathematical model.

Error Analysis

Error analysis involves comparing the outputs of a model with actual observed data to determine the accuracy and reliability of the model. This concept is important when assessing whether a model overestimates or underestimates real-world measurements, as seen when contrasting the predicted population with the data from a bar graph.

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