Americans are getting married later in life, or not getting...

Americans are getting married later in life, or not getting married at all. In 2010, more than half of Americans ages 25 through 29 were unmarried. The bar graph shows the percentage of never-married men and women in this age group for four selected years. The data are displayed as two sets of four points each, one scatter plot for the percentage of never-married American men and one for the percentage of never-married American women. Also shown for each scatter plot is a line that passes through or near the four points. Use these lines to solve Exercises 87–88. In this exercise, you will use the blue line for the women shown on the scatter plot to develop a model for the percentage of never-married American females ages $25-29$ a. Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of never-married American females ages $25-29, y, x$ years after 1980 b. Write the equation from part (a) in slope-intercept form. Use function notation. c. Use the linear function to predict the percentage of never married American females, ages $25-29,$ in 2020 .

Americans are getting married later in life, or not getting married at all. In 2010, more than half of Americans ages 25 through 29 were unmarried. The bar graph shows the percentage of never-married men and women in this age group for four selected years. The data are displayed as two sets of four points each, one scatter plot for the percentage of never-married American men and one for the percentage of never-married American women. Also shown for each scatter plot is a line that passes through or near the four points. Use these lines to solve Exercises 87–88.
In this exercise, you will use the blue line for the women shown on the scatter plot to develop a model for the percentage of never-married American females ages $25-29$
a. Use the two points whose coordinates are shown by the voice balloons to find the point-slope form of the equation of the line that models the percentage of never-married American females ages $25-29, y, x$ years after 1980
b. Write the equation from part (a) in slope-intercept form. Use function notation.
c. Use the linear function to predict the percentage of never married American females, ages $25-29,$ in 2020 .

Key Concepts

Function Notation

Function notation, such as f(x), is used to denote that the output value depends on the input x. In modeling scenarios, it provides a clear and concise way to represent the relationship between variables. This notation is especially useful in communicating predictions and models, as seen in the transition from an equation to a practical function for making future predictions.

Slope-Intercept Form

The slope-intercept form of a linear equation, represented as y = mx + b, explicitly shows both the slope (m) of the line and the y-intercept (b), making it easy to understand the rate of change and the starting value. This form is widely used for graphing and interpreting the behavior of a line within real-world contexts.

Extrapolation

Extrapolation is the process of using a linear model to predict values outside the range of the observed data. In this problem, extrapolation is used to estimate the percentage of never-married American females in a future year. It is an important concept in mathematics and statistics that allows for forecasting by extending the established trend, though it also comes with the caveat that the further one extrapolates, the less reliable the prediction may become.

Linear Modeling

Linear modeling involves constructing a line to approximate the relationship between two variables using observed data points. In the context of this problem, the linear model is used to describe how the percentage of never-married individuals changes over time. The concept is fundamental in algebra and statistics as it allows one to predict values beyond the observed data by assuming that the relationship is linear.

Point-Slope Form

The point-slope form of a linear equation, given by y - y? = m(x - x?), is a method used when a point on the line and the slope are known. This form is particularly useful for formulating a line that fits given data points and serves as a starting point for deriving other forms of linear equations, such as the slope-intercept form.

Transcript

00:01 So for this problem, we are going to be using the two points given to us to find the point slope form of the line for females that are never married.

00:11 So we're going to be using the equation of m equals y2 minus y1 over x2 minus x1.

00:21 And we're going to use this because we need to find point slope, which is y minus y1 equals x or equals m.

00:30 Parentheses x minus x1 so our two points that we're going to be using are 20 comma 38 .9 and 30 comma 47 .8 so if we label this point one and point two and all of the variables accordingly that's y1 actually so let's plug in and find m here because we don't know and it and give it to us.

01:04 So 47 .8 minus 38 .9 over 30 minus 20.

01:12 Well, 47 .8 minus 38 .9.

01:17 It's going to be, we do quick math here.

01:22 8 .9.

01:25 So that'll give us 8 .9 over 10, which is in reality, if you divide 8 .9 by 10, you get a slope of 0 .89.

01:38 So there is our m so now we can plug this in a two -point slope form by using our first equation so y minus 38 .9 is equal to 0 .89 of x x minus 20.

01:57 So that is how you find a the point slope form and now we have to convert this into slope -intercept form using function notation so in order to do that we need to distribute the point 8 to the parentheses and get y by itself so let's go ahead and do distribute first so y minus 38 .9 is equal to 0 .89 x minus 20 times 0 .89 well if we double 0 .89 then we should get 18 16 17 we double that then we get 1 .78 but we double that then we get 1 .78 but we're multiplying by 20 so it's actually going to be 17 .8 so 17 .8 and then we want to add 38 .9 to each side to get y by itself so we get y is equal to 0 .89x plus what's 38 .9 minus 17 .8 that's 1 and then that is also 1 and that's 2 so plus 21.

03:11 Point one...

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