Interpreting Slope in Equations and Graphs

Learning Outcomes

  • Solve slope applications using equations and graphs

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.  But 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
[latex]0[/latex] horizontal constant
undefined vertical N/A

Visual Interpretations of Slope

One of the ways that we can interpret slope is through a visual interpretation.  You have heard us describe slope as going “uphill” or “downhill”.  In the following application problems, we are interpreting slope visually in real-world contexts.

example

The pitch of a building’s roof is the slope of the roof. Knowing the pitch is important in climates where there is heavy snowfall. If the roof is too flat, the weight of the snow may cause it to collapse. What is the slope of the roof shown?

This figure shows a house with a sloped roof. The roof on one half of the building is labeled “pitch of the roof”. There is a line segment with arrows at each end measuring the vertical length of the roof and is labeled “rise = 9 feet”. There is a line segment with arrows at each end measuring the horizontal length of the root and is labeled “run = 18 feet”.

 

try it

 

example

Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.

Sewage pipes must slope down [latex]{\Large\frac{1}{4}}[/latex] inch per foot in order to drain properly. What is the required slope?

This figure shows a right triangle. The short leg is vertical and is labeled “1 over 4 inch”. The long leg labeled “1 foot”.

 

try it

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 [latex]1950[/latex] and [latex]2000[/latex] 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.

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

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

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 [latex](0, 25200)[/latex] [latex]\begin{array}{l}x_{1}=0\\y_{1}=25200\end{array}[/latex]
Point 2 [latex](50, 71400)[/latex] [latex]\begin{array}{l}x_{2}=50\\y_{2}=71400\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 [latex](0, 74400)[/latex] [latex]\begin{array}{l}x_{1}=1950\\y_{1}=74400\end{array}[/latex]
Point 2 [latex](50, 272700)[/latex] [latex]\begin{array}{l}x_{2}=2000\\y_{2}=272700\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 [latex]100[/latex], between [latex]2011[/latex] and [latex]2015[/latex] 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.

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

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

 

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 [latex]x[/latex] values represent years, and the y values represent the number of smokers. Remember that this dataset is per [latex]100[/latex] 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 [latex]1.75[/latex] per [latex]100[/latex] 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.