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Study Guides > ALGEBRA / TRIG I

Interpreting the y-Intercept and Making Predictions

Learning Outcomes

  • Interpret the characteristics of a linear equation and use that equation to make predictions

Interpret the y-intercept of a linear equation

Earlier in this module, we learned how to write the equation of a line given the slope and yy-intercept.  Often, when the line in question represents a set of data or observations, the yy-intercept can be interpreted as a starting point.  We will continue to use the examples for house value in Mississippi and Hawaii and high school smokers to interpret the meaning of the yy-intercept in those equations.

Example

Recall the equations and data for house value: Linear equations describing the change in median home values between 19501950 and 20002000 in Mississippi and Hawaii are as follows: Hawaii:  y=3966x+74,400y = 3966x+74,400 Mississippi:  y=924x+25,200y = 924x+25,200 The equations are based on the following dataset. xx = the number of years since 19501950, and y = the median value of a house in the given state.
Year (x) Mississippi House Value (y) Hawaii House Value (y)
00 $25,200 $74,400
5050 $71,400 $272,700 
And the equations and data for high school smokers: A linear equation describing the change in the number of high school students who smoke, in a group of 100100, between 20112011 and 20152015 is given as:

 y=1.75x+16y = -1.75x+16

And is based on the data from this table, provided by the Centers for Disease Control. xx = the number of years since 20112011, and yy = the number of high school smokers per 100100 students.
Year Number of  High School Students Smoking Cigarettes (per 100)
00 1616
44 99
Also recall that the equation of a line in slope-intercept form is as follows:

y=mx+by = mx + b

     m    =   slope(x,y)=   a point on the line       b    =   the y value of the y-intercept\begin{array}{l}\,\,\,\,\,m\,\,\,\,=\,\,\,\text{slope}\\(x,y)=\,\,\,\text{a point on the line}\\\,\,\,\,\,\,\,b\,\,\,\,=\,\,\,\text{the y value of the y-intercept}\end{array}

 

The examples that follow show how to interpret the y-intercept of the equations used to model house value and the number of high school smokers. Additionally, you will see how to use the equations to make predictions about house value and the number of smokers in future years.

Example

Interpret the y-intercepts of the equations that represent the change in house value for Hawaii and Mississippi. Hawaii:  y=3966x+74,400y = 3966x+74,400 Mississippi:  y=924x+25,200y = 924x+25,200 The y-intercept of a two-variable linear equation can be found by substituting 00 in for xx.

Hawaii

y=3966x+74,400y=3966(0)+74,400y=74,400y = 3966x+74,400\\y = 3966(0)+74,400\\y = 74,400

The yy-intercept is a point, so we write it as (0, 74,400).  Remember that yy-values represent dollars and xx values represent years.  When the year is 00—in this case 00 because that is the first date we have in the dataset—the price of a house in Hawaii was $74,400.  (Remember that xx represents the number of years since 19501950, so if x=0x=0 the year is 19501950.)

Mississippi

y=924x+25,200y=924(0)+25,200y=25,200y = 924x+25,200\\y = 924(0)+25,200\\y = 25,200

The yy-intercept is (0,25,200)(0, 25,200).  This means that in 19501950 the value of a house in Mississippi was $25,200.

Example

Interpret the yy-intercept of the equation that represents the change in the number of high school students who smoke out of 100100. Substitute 00 in for xx.

y=1.75x+16y=1.75(0)+16y=16y = -1.75x+16\\y = -1.75(0)+16\\y = 16

The yy-intercept is (0,16)(0,16).  The data starts at 20112011, so we represent that year as 00. We can interpret the yy-intercept as follows: In the year 20112011, 1616 out of every 100100 high school students smoked.

In the following video you will see an example of how to interpret the yy- intercept given a linear equation that represents a set of data.

https://youtu.be/Yhtl28DRqfU

Use a linear equation to make a prediction

Another useful outcome we gain from writing equations from data is the ability to make predictions about what may happen in the future. We will continue our analysis of the house price and high school smokers. In the following examples you will be shown how to predict future outcomes based on the linear equations that model current behavior.

Example

Use the equations for house value in Hawaii and Mississippi to predict house value in 20352035. We are asked to find house value, y, when the year, xx, is 20352035. Since the equations we have represent house value increase since 19501950, we have to be careful. We can't just plug in 20352035 for xx, because xx represents the years since 19501950. How many years are between 19501950 and 20352035? 20351950=852035 - 1950 = 85 This is our xx-value. For Hawaii:

y=3966x+74,400y=3966(85)+74,400y=337110+74,400=411,510y = 3966x+74,400\\y = 3966(85)+74,400\\y = 337110+74,400 = 411,510

Holy cow! The average price for a house in Hawaii in 20352035 is predicted to be $411,510 according to this model. See if you can find the current average value of a house in Hawaii. Does the model measure up? For Mississippi:

y=924x+25,200y=924(85)+25,200y=78540+25,200=103,740y = 924x+25,200\\y = 924(85)+25,200\\y = 78540+25,200 = 103,740

The average price for a home in Mississippi in 20352035 is predicted to be $103,740 according to the model. See if you can find the current average value of a house in Mississippi. Does the model measure up?
In the following video, you will see the example of how to make a prediction with the home value data. https://youtu.be/Bw9XjDAl-K0

Try It

[ohm_question]188447[/ohm_question]
 

Example

Use the equation for the number of high school smokers per 100100 to predict the year when there will be 00 smokers per 100100.

y=1.75x+16y = -1.75x+16

This question takes a little more thinking.  In terms of xx and yy, what does it mean to have 00 smokers?  Since y represents the number of smokers and xx represent the year, we are being asked when yy will be 00. Substitute 00 for yy.

y=1.75x+16y = -1.75x+16

0=1.75x+160 = -1.75x+16

16=1.75x-16 = -1.75x

161.75=x\frac{-16}{-1.75} = x

x=9.14x = 9.14 years

Again, like the last example, xx is representing the number of years since the start of the data—which was 20112011, based on the table:
Year Number of  High School Students Smoking Cigarettes (per 100)
00 1616
44 99
So we are predicting that there will be no smokers in high school by 2011+9.14=20202011+9.14=2020. How accurate do you think this model is? Do you think there will ever be 00 smokers in high school?
The following video gives a thorough explanation of making a prediction given a linear equation. https://youtu.be/5W0qq8saxO0  

Bringing it Together

The last example we will show will include all of the concepts that we have been building up throughout this module.  We will interpret a word problem, write a linear equation from it, graph the equation, interpret the yy-intercept and make a prediction. Hopefully this example will help you to make connections between the concepts we have presented.

Example

It costs $600 to purchase an iphone, plus $55 per month for unlimited use and data. Write a linear equation that represents the cost, yy,  of owning and using the iPhone for xx amount of months. When you have written your equation, answer the following questions:
  1. What is the total cost you’ve paid after owning and using your phone for 2424 months?
  2. If you have spent $2,580 since you purchased your phone, how many months have you used your phone?
5 iPhones laying next to each other iPhone
[hidden-answer a="282349"][/hidden-answer] Read and Understand: We need to write a linear equation that represents the cost of owning and using an iPhone for any number of months.  We are to use yy to represent cost, and xx to represent the number of months we have used the phone. Define and Translate: We will use the slope-intercept form of a line, y=mx+by=mx+b, because we are given a starting cost and a monthly cost for use.  We will need to find the slope and the yy-intercept. Slope: in this case we don't know two points, but we are given a rate in dollars for monthly use of the phone.  Our units are dollars per month because slope is ΔyΔx\frac{\Delta{y}}{\Delta{x}}, and yy is in dollars and xx is in months. The slope will be 55 dollars 1 month \frac{55\text{ dollars }}{1\text{ month }}m=551=55m=\frac{55}{1}=55 Y[/latex]-Intercept: the [latex]y[/latex]-intercept is defined as a point [latex]\left(0,b\right)[/latex].  We want to know how much money we have spent, [latex]y[/latex], after [latex]0[/latex] months.  We haven't paid for service yet, but we have paid [latex]$600[/latex] for the phone. The [latex]y[/latex]-intercept in this case is called an initial cost. [latex]b=600 Write and Solve: Substitute the slope and intercept you defined into the slope-intercept equation.

y=mx+by=55x+600\begin{array}{c}y=mx+b\\y=55x+600\end{array}

Now we will answer the following questions:

  1. What is the total cost you’ve paid after owning and using your phone for 2424 months?
Since xx represents the number of months you have used the phone, we can substitute x=24x=24 into our equation.

y=55x+600y=55(24)+600y=1320+600y=1920\begin{array}{c}y=55x+600\\y=55\left(24\right)+600\\y=1320+600\\y=1920\end{array}

YY represents the cost after xx number of months, so in this scenario, after 2424 months, you have spent $1920 to own and use an iPhone.

  1. If you have spent $2,580 since you purchased your phone, how many months have you used your phone?
We know that yy represents cost, and we are given a cost and asked to find the number of months related to having spent that much. We will substitute y=$2,580 into the equation, then use what we know about solving linear equations to isolate xx:

 \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,y=55x+600\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2580=55x+600\\\text{ subtract 600 from each side}\,\,\,\,\,\,\,\underline{-600}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\underline{-600}\\\text{}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1980=55x\\\text{}\\\text{ divide each side by 55 }\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{1980}{55}=\frac{55x}{55}\\\text{}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,36=x\end{array}

If you have spent $2,580 then you have been using your iPhone for 3636 months, or 33 years.

 

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