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 difference between the female U.S. population and the male U.S. population for the years shown in the bar graph. b. Use the function from part (a) to find how many more women than men there were in the U.S. population in 2010 c. Does the result in part (b) overestimate or underestimate the actual difference between the female and male population in 2010 shown by the bar graph? By how much?

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 difference between the female U.S. population and the male U.S. population for the years shown in the bar graph.
b. Use the function from part (a) to find how many more women than men there were in the U.S. population in 2010
c. Does the result in part (b) overestimate or underestimate the actual difference between the female and male population in 2010 shown by the bar graph? By how much?

Key Concepts

Model Error Analysis

Model error analysis involves comparing the predictions made by mathematical models to actual observed data from sources like graphs or historical records. This concept helps in assessing how well the model approximates reality by identifying overestimates or underestimates, and quantifying the error to improve future models or understand the limitations of the current approach.

Function Difference

The difference between two functions, obtained by subtracting one from the other, creates a new function that represents the net difference between the two original quantities. This concept is useful when comparing two models, such as the populations of different groups, to derive insights about their relative sizes or trends over time.

Linear Functions

Linear functions are mathematical models that represent a constant rate of change and are typically expressed in the form y = mx + b, where m is the slope and b is the y-intercept. They are widely used to model relationships and trends in data over time, such as population growth, by providing a simple approximation of how one variable changes with respect to another.

Slope-Intercept Form

The slope-intercept form is a straightforward way to express linear functions. It clearly shows the rate of change (slope) and the initial value (y-intercept), making it easy to understand and compare different trends. This form is particularly useful in applications such as population modeling, where the rate of change directly relates to growth or decline over time.

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