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 scater plot is a line that passes through or near the four points Use these lines to solve (figure can't copy) In this exercise, you will use the red line for the men shown on the scatter plot to develop a model for the percentage of never-married American males 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 males 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 nevermarried American males, ages $25-29,$ in 2015 .

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 scater plot is a line that passes through or near the four points Use these lines to solve (figure can't copy)
In this exercise, you will use the red line for the men shown on the scatter plot to develop a model for the percentage of never-married American males 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 males 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 nevermarried American males, ages $25-29,$ in 2015 .

Key Concepts

Linear Function Modeling

This concept involves creating a mathematical representation using a linear equation to approximate a relationship between two variables. It is useful in contexts where data points can be modeled as a straight line, representing a constant rate of change over time.

Point-Slope Form

Point-slope form is a way to write the equation of a line when you know one point on the line and the slope. Expressed as y - y? = m(x - x?), it directly ties the rate of change (slope) to a particular point, making it useful for linear modeling based on data.

Slope-Intercept Form

Slope-intercept form expresses a line as y = mx + b, where m is the slope and b is the y-intercept. This form is widely used because it clearly shows how changes in the independent variable x affect the dependent variable y and immediately reveals the starting value when x is zero.

Function Notation

Function notation, typically written as f(x), succinctly expresses the relationship between an input and its corresponding output in a function. It provides a standard way to denote and evaluate mathematical models, including linear functions, making it easier to communicate and use the relationships.

Prediction Using Linear Models

This concept involves substituting a specific input value into the linear model to estimate the corresponding output. It is a common application of linear functions in forecasting future values based on observed trends in historical data.

Transcript

00:03 Okay, for part 8, they give you the two points that you use.

00:07 And i started off by labeling mine, x1, y, 1, x2, y, 2, y, 2, so the numbers are ready to go.

00:16 And they've given you, they've given me points.

00:19 So you have those, but you don't want to slope.

00:21 So you want to find the slope between these two points.

00:24 You're going to use the slope formula, y2 minus y, 1, over x2 minus x1.

00:29 You just plug these numbers in.

00:31 You've got 62 .6 over.

00:34 I'm sorry, 62 .6 minus y1.

00:37 5 to 1 .7 x2 is 30 x1 is 20 and you should get if you calculate it 1 .09 for your slip that's just subtracting and then dividing okay so this is your m and when you could put it in point slope form you can use the point slope formula y 2 minus 1 equals x times x minus sorry m times x minus x1 and this is your point slope and this is your point.

01:15 Slope form.

01:17 It's going to abbreviate.

01:22 You can use either of these two points from the beginning.

01:27 So this one.

01:28 I use this one because it already has it labeled x1 .y one, but you can easily use the second one just the same.

01:36 Doesn't matter.

01:39 So using the first point would be y minus 51 .7 equals ym is 1 .09 x minus x1 which is 20.

01:52 And then you're going to distribute get this guy right here, the 1 .09.

01:57 So he's the same over here, 3 .7.

02:01 He turns into 109x minus 21 .8...