Applications: Interpreting Slope in Equations and Graphs

Learning Objectives

  • Interpret slope in equations and graphs
    • Verify the slope of a linear equation given a dataset
    • Interpret the slope of a linear equation as it applies to a real situation

Verify Slope From a Dataset

Massive amounts of data is being collected every day by a wide range of institutions and groups.  This data is used for many purposes including business decisions about location and marketing, government decisions about allocation of resources and infrastructure, and personal decisions about where to live or where to buy food.

In the following example, you will see how a dataset can be used to define the slope of a linear equation.

Example

Given the dataset, verify the values of the slopes of each equation.

Linear equations describing the change in median home values between 1950 and 2000 in Mississippi and Hawaii are as follows:

Hawaii:  [latex]y=3966x+74,400[/latex]

Mississippi:  [latex]y=924x+25,200[/latex]

The equations are based on the following dataset.

x = the number of years since 1950, and y = the median value of a house in the given state.

Year (x) Mississippi House Value (y) Hawaii House Value (y)
0 $25,200 $74,400
50 $71,400 $272,700  

The slopes of each equation can be calculated with the formula you learned in the section on slope.

[latex]\displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}[/latex]

Mississippi:

Name Ordered Pair Coordinates
Point 1 (0, 25,200) [latex]\begin{array}{l}x_{1}=0\\y_{1}=25,200\end{array}[/latex]
Point 2 (50, 71,400) [latex]\begin{array}{l}x_{2}=50\\y_{2}=71,400\end{array}[/latex]

[latex]\displaystyle m=\frac{{71,400}-{25,200}}{{50}-{0}}=\frac{{46,200}}{{50}} = 924[/latex]

We have verified that the slope [latex]\displaystyle m = 924[/latex] matches the dataset provided.

Hawaii:

Name Ordered Pair Coordinates
Point 1 (0, 74,400) [latex]\begin{array}{l}x_{1}=1950\\y_{1}=74,400\end{array}[/latex]
Point 2 (50, 272,700) [latex]\begin{array}{l}x_{2}=2000\\y_{2}=272,700\end{array}[/latex]

[latex]\displaystyle m=\frac{{272,700}-{74,400}}{{50}-{0}}=\frac{{198,300}}{{50}} = 3966[/latex]

We have verified that the slope [latex]\displaystyle m = 3966[/latex] matches the dataset provided.

Example

Given the dataset, verify the values of the slopes of the equation.

A linear equation describing the change in the number of high school students who smoke, in a group of 100, between 2011 and 2015 is given as:

 [latex]y = -1.75x+16[/latex]

And is based on the data from this table, provided by the Centers for Disease Control.

x = the number of years since 2011, and y = the number of high school smokers per 100 students.

Year Number of  High School Students Smoking Cigarettes (per 100)
0 16
4 9
Name Ordered Pair Coordinates
Point 1 (0, 16) [latex]\begin{array}{l}x_{1}=0\\y_{1}=16\end{array}[/latex]
Point 2 (4, 9) [latex]\begin{array}{l}x_{2}=4\\y_{2}=9\end{array}[/latex]

[latex]\displaystyle m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{{9-16}}{{4-0}} =\frac{{-7}}{{4}}=-1.75[/latex]

We have verified that the slope [latex]\displaystyle{m=-1.75}[/latex] matches the dataset provided.

Interpret the Slope of  Linear Equation

Okay, now we have verified that data can provide us with the slope of a linear equation. So what? We can use this information to describe how something changes using words.

First, let’s review the different kinds of slopes possible in a linear equation.

Uphill line with positive slope has a line that starts at the bottom-left and goes into the top-right of the graph. Downhill line with negative slope starts in the top-left and ends in the bottom-right part of the graph. Horizontal lines have a slope of 0. Vertical lines have an undefined slope.

We often use specific words to describe the different types of slopes when we are using lines and equations to represent “real” situations. The following table pairs the type of slope with the common language used to describe it both verbally and visually.

Type of Slope Visual Description  Verbal Description
positive uphill increasing
negative downhill decreasing
0 horizontal constant
undefined vertical N/A

Example

Interpret the slope of each equation for house values using words.

Hawaii:  [latex]y = 3966x+74,400[/latex]

Mississippi:  [latex]y = 924x+25,200[/latex]

Interpret the Meaning of the Slope Given a Linear Equation—Median Home Values

Example

Interpret the slope of the line describing the change in the number of high school smokers using words.

Apply units to the formula for slope. The x values represent years, and the y values represent the number of smokers. Remember that this dataset is per 100 high school students.

[latex]\displaystyle m=\frac{{9-16}}{{2015-2011}} =\frac{{-7 \text{ smokers}}}{{4\text{ year}}}=-1.75\frac{\text{ smokers}}{\text{ year}}[/latex]

The slope of this linear equation is negative, so this tells us that there is a decrease in the number of high school age smokers each year.

The number of high schoolers that smoke decreases by 1.75 per 100 each year.

Interpret the Meaning of the Slope of a Linear Equation—Smokers

On the next page, we will see how to interpret the y-intercept of a linear equation, and make a prediction based on a linear equation.