Section 2.5 - Applications of Linear Equations

Section Objectives

1. Construct a scatterplot.
2. Recognize a linear relationship, and determine an equation that describes the relationship.
3. Use an equation to make predictions in a linear relationship.
4. Compute an average rate of change.

Scatterplots and Linear Trends

A scatterplot is a collection of plotted points obtained from ordered pairs. In a scatterplot, the plotted points are not connected with a curve.

We often use scatterplots to identify patterns and recognize trends. If plotted points appear to approximately "fit" a line, we can use a linear equation to describe the pattern and predict behavior. Finding a "best fit" line is a very important practical problem.

There are well-defined procedures for finding best fit lines. For the most part, we will use graphs to estimate best fit lines.

Example

• The following data describe a random sample of used Porsche sports cars from Autotrader.com.

  Mileage (in thousands), x	Price (in $1000s), y
  16.7	95.8
  73.2	30.1
  32.7	45.7
  18.2	91.7
  23.4	79.5
  54.1	41.0
  41.9	65.0
  26.2	72.0

  1.) Construct the corresponding scatterplot. Use mileage for your horizontal axis and price for your vertical axis.

  
  
  
  

  2.) Draw the line through your collection of points that "best fits" the data.

  
  
  
  

  3.) Approximate the slope and $y$-intercept of your best-fit line. Then write the equation of your line in slope-intercept form.

  
  
  
  

  4.) Based on your linear model, how much should a Porsche sports car cost if it has 60,000 miles?

  
  
  
  

  5.) Based on your linear model, about how many miles would you expect a $55,000 Porsche to have?

  
  
  
  

  6.) Use the data points in the table corresponding to the least and most miles to compute the average rate of change of cost per mile, $\Delta y/\Delta x$. How does this compare with your estimated slope?

  
  
  
  

  7.) Use your equation from #3 with $x=16.7$ and $x=73.2$ to compute $\Delta y/\Delta x$. How does this compare with your estimated slope?