Linear Functions and Graphs

1. Calculating and Interpreting Slope

As you may recall from previous coursework, the slope of a line between two points is which is often represented as

Consider the graph shown. Given two points on a line, which we call and we calculate the slope as follows.

formula to know

Slope

The slope of a line containing the points and is given by
Using function notation, where the slope can also be written

EXAMPLE

Consider the points and



To find the slope of the line containing the points, write and then use the slope formula.

The slope of the line containing these points is

This is depicted on the graph. To get from the left-hand point to the right-hand point, you move up 3 units, then move right 4 units.

Note that this graph describes y as a function of x.

EXAMPLE

Consider the points and



To find the slope of the line containing the points, write and then use the slope formula.

The slope of the line containing these points is

This is depicted on the graph. To get from to move down 2 units, then to the right 6 units. This translates to

Note that this graph describes y as a function of x.

EXAMPLE

Consider the points and



To find the slope, write and then use the slope formula.

The slope of the line containing these points is

This is depicted on the graph. Moving from left to right, there is no rise over a horizontal run of 6 units.

Note that this graph describes y as a function of x.

watch

In this video, we will find the slope of a line given the graph of the line.

Note that the lines in all three examples were functions. Based on this simple fact, we define a linear function as a function whose graph is a line.

big idea

Given a linear function, we make the following connections about slope:

• A positive slope indicates that the line is rising to the right, indicating that the function is increasing.
• A negative slope indicates that the line is falling to the right, indicating that the function is decreasing.
• A zero slope indicates that the line is flat, indicating that the function is constant.

Here is a problem for you to try.

try it

Consider the line that contains the points and

Calculate the slope of the line.

Using the formula the slope is:

Is the linear function that contains these points increasing or decreasing?

It is decreasing since the slope is negative.

When x and y represent quantities, the slope can be interpreted as a rate of change.

EXAMPLE

Suppose that in 2017, the population of a small town was 4,124 and in 2021, the population rose to 5,000.

Question: What was the average population growth over the four-year period?

The population increased from 4,124 to 5,000, which is a difference of 876 people. Then, the rate at which the population changed is

Notice how slope can be used to answer this question. If we let the population of the town and the year, then we have two ordered pairs to describe the situation: and

Then, the slope of the line that joins these two points is:

Thus, the population increased on average by 219 people per year.

big idea

• The slope can be interpreted as a rate of change.
• If we view x as the input and y as the output (as with a function), then the units of slope are

Here is a problem for you to try on your own.

try it

Jessica is walking home from a friend’s house. Two minutes after leaving, she is 1.4 miles from home. Twelve minutes after leaving, she is 0.9 miles from home.

What is her average rate in miles per hour?

The time frame we are working with is minutes.

The distance traveled in this time frame is miles.

Then, Jessica’s average rate of change is

To convert to miler per hour, remember that there are 60 minutes in every hour.

Then, her rate is

Now that we have made the connection between slope and linear functions, let’s take a closer look at how these ideas intertwine.

term to know

Linear Function

A function whose graph is a line.

2. Representing Linear Functions

There are four forms that are used to represent a function: word form, equation (function) form, table form, and graph form. Through this tutorial, we will see how these are connected when the situation is modeled by a linear function.

A key connection to make is between the equation of the linear function, its corresponding table of values, and its graph.

Consider the equation

Here is a table of values:

	0	1	2	3	4
	3	5	7	9	11

Here is the graph of this function:

Note that is a linear function, and therefore must have a slope.

Let’s take two input-output pairs: and

Calculate the slope between them:

Let’s choose two other points: and

Calculate the slope between them:

In fact, given that the graph is a line, the slope of the line between any two points should be 2.

Notice also that when This means that the graph has y-intercept

Given the equation of any linear function, we can make some generalizations.

big idea

The equation of a linear function is or using function notation,

The value of m is the slope and represents the constant rate of change.

The value of b is the y-coordinate of the y-intercept. That is, the point is on the line.

You may remember from previous coursework that the equation has a special name, called the slope-intercept form.

The slope-intercept form of a line is the most convenient to use to model situations. Through the next few examples, we will see how the graph, table, function, and verbal descriptions relate to each other.

EXAMPLE

John is mountain climbing and begins his journey at an elevation of 500 feet. He also ascends at a rate of 80 feet per hour.

Let the number of hours that John is ascending, and his elevation after t hours. Since John is ascending at a constant rate, a linear function can be used to model this situation.

Let’s now look at the four ways to represent this function:

Word form:
John’s initial elevation is 500 feet and increases by 80 feet every hour.

Table form:
Using whole hours as inputs, here is a table:

hours	0	1	2	3	4	5
elevation (feet)	500	580	660	740	820	900

Note how the value of increases by 80 units for every unit that t increases.

Equation form:
Normally, we seek a model of the form Since t is the input and is the output, our model will have the form

To write an equation, note that John’s elevation when is 500 feet, indicating that this is the y-intercept. From the table above, the constant rate of change is 80 feet per hour, indicating that the slope of the model is 80.

At this point, we deduce that

Graph form:
Using the points from the table, we have the following graph. Note that the graph rises to the right, indicating that is an increasing function.



Note that only nonnegative values of t are used in this situation. This means the domain of this function is Even though the expression for is defined for all possible values of t, only nonnegative values of t will provide meaningful results. This is because John can’t climb mountains for a negative amount of time.

Now that we have this insight, writing the equation of a linear function will be a bit easier.

EXAMPLE

A basement has flooded. After pumping begins, the amount of water (in gallons) in the basement t hours after pumping began is given by the function

Let’s find and interpret the other forms of the function. Since the equation was given, the table and graph follow nicely.

Equation form:

Table form:

hours	0	1	2	3	4	5
gallons remaining	8000	6800	5600	4400	3200	2000

Notice: Every time t increases by one unit, the value of decreases by 1200 units.

Graph form:
Note that the graph falls to the right, which means is decreasing.



Notice that the number of gallons in the basement is decreasing over time. Once there are zero gallons remaining, the pumping will stop. We can use the equation to find when this happens:

	Set to 0.
	Replace with
	Subtract 8000 from both sides.
	Divide both sides by -1200.
	Simplify.

This means that the basement is empty after hours. Or 6 hours and 40 minutes. This is represented on the graph by the t-intercept.

Word Form:
Initially, there are 8000 gallons of water in the basement. After pumping begins, the water is removed at a constant rate of 1200 gallons per hour.

We have seen examples with rates of change that led to increasing and decreasing functions. Now, let’s look at one where the function is constant.

EXAMPLE

A cellular phone plan costs $80 regardless of how much data is used. Let the cost of the plan after using x gigabytes of data.

Here is a table of values for various input-output pairs:

	0	1	2	3	6	20
	80	80	80	80	80	80

Regardless of how much data is used, the cost remains the same.

Written in equation form, Regardless of the input, the output is 80, which represents a plan cost of $80.

The graph of the function is below. Note that the graph is not rising nor falling, indicating that is a constant function.

The equation in the last example was written To align with the slope-intercept form, it could also be written This means that the slope of the line is 0. Through the last three examples, this leads to an important connection between the slope of the line and whether its corresponding function is increasing, decreasing, or constant.

big idea

Given that a situation is modeled by a linear function

• A positive slope indicates that is increasing.
• A negative slope indicates that is decreasing.
• A zero slope indicates that is constant.

try it

A boat is 100 miles from the marina and is sailing toward it at a rate of 10 miles per hour.

Write the equation D   (t  ), the distance that the boat is from the marina after t hours.

Here is an example that is very applicable to everyday life: converting between Celsius and Fahrenheit.

try it

Given a temperature measured in degrees Celsius, the temperature in degrees Fahrenheit is given by the function

If the temperature increases by 1° Celsius, does the Fahrenheit temperature increase or decrease, and by how much?

The Fahrenheit temperature increases by 1.8 degrees.

For a temperature of 20° Celsius, what is the corresponding temperature in degrees Fahrenheit?

68° F

term to know

Slope-Intercept Form

A linear equation in the form where m is the slope (constant rate of change) and is the y-intercept.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.