# 3.2: Linear Functions


##### Learning Objectives
• Represent a linear function.
• Determine whether a linear function is increasing, decreasing, or constant.
• Interpret slope as a rate of change.
• Write and interpret an equation for a linear function.
• Graph linear functions.
• Determine whether lines are parallel or perpendicular.
• Write the equation of a line parallel or perpendicular to a given line.

Just as with the growth of a bamboo plant, there are many situations that involve constant change over time. Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train (maglev train in the world, the Shanghai MagLev Train (Figure $$\PageIndex{1}$$). It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minutes.

Suppose a maglev train were to travel a long distance, and that the train maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.maglev train were to travel a long distance, and that the train maintains a constant speed of 83 meters per second for a period of time once it is 250 meters from the station. How can we analyze the train’s distance from the station as a function of time? In this section, we will investigate a kind of function that is useful for this purpose, and use it to investigate real-world situations such as the train’s distance from the station at a given point in time.

## Representing Linear Functions

The function describing the train’s motion is a linear function, which is defined as a function with a constant rate of change, that is, a polynomial of degree 1. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. We will describe the train’s motion as a function using each method.

### Representing a Linear Function in Word Form

Let’s begin by describing the linear function in words. For the train problem we just considered, the following word sentence may be used to describe the function relationship.

• The train’s distance from the station is a function of the time during which the train moves at a constant speed plus its original distance from the station when it began moving at constant speed.

The speed is the rate of change. Recall that a rate of change is a measure of how quickly the dependent variable changes with respect to the independent variable. The rate of change for this example is constant, which means that it is the same for each input value. As the time (input) increases by 1 second, the corresponding distance (output) increases by 83 meters. The train began moving at this constant speed at a distance of 250 meters from the station.

### Representing a Linear Function in Function Notation

Another approach to representing linear functions is by using function notation. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where xis the input value, $$m$$ is the rate of change, and $$b$$ is the initial value of the dependent variable.

\begin{align*} &\text{Equation form } &y=mx+b \\[4pt] &\text{Equation notation } &f(x)=mx+b \end{align*}

In the example of the train, we might use the notation $$D(t)$$ in which the total distance $$D$$ is a function of the time $$t$$. The rate, $$m$$, is 83 meters per second. The initial value of the dependent variable $$b$$ is the original distance from the station, 250 meters. We can write a generalized equation to represent the motion of the train.

$D(t)=83t+250$

### Representing a Linear Function in Tabular Form

A third method of representing a linear function is through the use of a table. The relationship between the distance from the station and the time is represented in Figure $$\PageIndex{2}$$. From the table, we can see that the distance changes by 83 meters for every 1 second increase in time.

##### Q/A? Can the input in the previous example be any real number?

No. The input represents time, so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. The input consists of non-negative real numbers.

### Representing a Linear Function in Graphical Form

Another way to represent linear functions is visually, using a graph. We can use the function relationship from above, $$D(t)=83t+250$$, to draw a graph, represented in Figure $$\PageIndex{3}$$. Notice the graph is a line. When we plot a linear function, the graph is always a line.

The rate of change, which is constant, determines the slant, or slope of the line. The point at which the input value is zero is the vertical intercept, or y-intercept, of the line. We can see from the graph in Figure $$\PageIndex{3}$$ that the y-intercept in the train example we just saw is $$(0,250)$$ and represents the distance of the train from the station when it began moving at a constant speed.

Notice that the graph of the train example is restricted, but this is not always the case. Consider the graph of the line $$f(x)=2x+1$$. Ask yourself what numbers can be input to the function, that is, what is the domain of the function? The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product.

##### Definition: Linear Function

A linear function is a function whose graph is a line. Linear functions can be written in the slope-intercept form of a line

$f(x)=mx+b$

where $$b$$ is the initial or starting value of the function (when input, $$x=0$$), and $$m$$ is the constant rate of change, or slope of the function. The y-intercept is at $$(0,b)$$.

##### Example $$\PageIndex{1}$$: Using a Linear Function to Find the Pressure on a Diver

The pressure, $$P$$, in pounds per square inch (PSI) on the diver in Figure $$\PageIndex{4}$$ depends upon her depth below the water surface, $$d$$, in feet. This relationship may be modeled by the equation, $$P(d)=0.434d+14.696$$. Restate this function in words.

To restate the function in words, we need to describe each part of the equation. The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths.

Analysis

The initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. The rate of change, or slope, is 0.434 PSI per foot. This tells us that the pressure on the diver increases 0.434 PSI for each foot her depth increases.

## Determining whether a Linear Function Is Increasing, Decreasing, or Constant

The linear functions we used in the two previous examples increased over time, but not every linear function does. A linear function may be increasing, decreasing, or constant. For an increasing function, as with the train example, the output values increase as the input values increase. The graph of an increasing function has a positive slope. A line with a positive slope slants upward from left to right as in Figure $$\PageIndex{5}$$(a). For a decreasing function, the slope is negative. The output values decrease as the input values increase. A line with a negative slope slants downward from left to right as in Figure $$\PageIndex{5}$$(b). If the function is constant, the output values are the same for all input values so the slope is zero. A line with a slope of zero is horizontal as in Figure $$\PageIndex{5}$$(c).

Increasing and Decreasing Functions

The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function.

• $$f(x)=mx+b$$ is an increasing function if $$m>0$$.
• $$f(x)=mx+b$$ is an decreasing function if $$m<0$$.
• $$f(x)=mx+b$$ is a constant function if $$m=0$$.
##### Example $$\PageIndex{2}$$: Deciding whether a Function Is Increasing, Decreasing, or Constant

Some recent studies suggest that a teenager sends an average of 60 texts per day. For each of the following scenarios, find the linear function that describes the relationship between the input value and the output value. Then, determine whether the graph of the function is increasing, decreasing, or constant.

1. The total number of texts a teen sends is considered a function of time in days. The input is the number of days, and output is the total number of texts sent.
2. A teen has a limit of 500 texts per month in his or her data plan. The input is the number of days, and output is the total number of texts remaining for the month.

##### Example $$\PageIndex{9}$$: Writing an Equation for a Linear Function Given Two Points

If $$f$$ is a linear function, with $$f(3)=−2$$, and $$f(8)=1$$, find an equation for the function in slope-intercept form.

Solution

We can write the given points using coordinates.

\begin{align*} f(3)&= -2{\rightarrow}(3,2) \\ f(8)&=1{\rightarrow}(8,1) \end{align*}

We can then use the points to calculate the slope.

\begin{align*} m&=\dfrac{y_2-y_1}{x_2-x_1} \\ &=\dfrac{1-(-2)}{8-3} \\ &=\dfrac{3}{5} \end{align*}

Substitute the slope and the coordinates of one of the points into the point-slope form.

\begin{align*} y-y_1&=m(x-x_1) \\ y-(-2)&=\dfrac{3}{5}(x-3) \end{align*}

We can use algebra to rewrite the equation in the slope-intercept form.

\begin{align*} y+2&=\dfrac{3}{5}(x-3) \\ y+2&=\dfrac{3}{5}x-\dfrac{9}{5} \\ y&=\dfrac{3}{5}x-\dfrac{19}{5} \end{align*}

##### Tri It! $$\PageIndex{1}$$

If $$f(x)$$ is a linear function, with $$f(2)=–11$$, and $$f(4)=−25$$, find an equation for the function in slope-intercept form.

$$y=−7x+3$$

## Modeling Real-World Problems with Linear Functions

In the real world, problems are not always explicitly stated in terms of a function or represented with a graph. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. As long as we know, or can figure out, the initial value and the rate of change of a linear function, we can solve many different kinds of real-world problems.

Given a linear function $$f$$ and the initial value and rate of change, evaluate $$f(c)$$.

1. Determine the initial value and the rate of change (slope).
2. Substitute the values into $$f(x)=mx+b$$.
3. Evaluate the function at $$x=c$$.
##### Example $$\PageIndex{10}$$: Using a Linear Function to Determine the number of Songs in a Music Collection

Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, $$N$$, in his collection as a function of time, $$t$$, the number of months. How many songs will he own in a year?

Solution

The initial value for this function is 200 because he currently owns 200 songs, so $$N(0)=200$$, which means that $$b=200$$.

The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that $$m=15$$. We can substitute the initial value and the rate of change into the slope-intercept form of a line.

We can write the formula $$N(t)=15t+200$$.

With this formula, we can then predict how many songs Marcus will have in 1 year (12 months). In other words, we can evaluate the function at $$t=12$$.

\begin{align*} N(12)&=15(12)+200 \\ &=180+200 \\ &= 380 \end{align*}

Marcus will have 380 songs in 12 months.

Analysis

Notice that $$N$$ is an increasing linear function. As the input (the number of months) increases, the output (number of songs) increases as well.

##### Example $$\PageIndex{11}$$: Using a Linear Function to Calculate Salary Plus Commission

Working as an insurance salesperson, Ilya earns a base salary plus a commission on each new policy. Therefore, Ilya’s weekly income, I,depends on the number of new policies, $$n$$, he sells during the week. Last week he sold 3 new policies, and earned $760 for the week. The week before, he sold 5 new policies and earned$920. Find an equation for $$I(n)$$, and interpret the meaning of the components of the equation.

Solution

The given information gives us two input-output pairs: $$(3,760)$$ and $$(5,920)$$. We start by finding the rate of change.

\begin{align*} m&=\dfrac{920-760}{5-3} \\ &=\dfrac{160}{2 \text{ policies}} \\ &=80 \text{ per policy} \end{align*}

Keeping track of units can help us interpret this quantity. Income increased by $160 when the number of policies increased by 2, so the rate of change is$80 per policy. Therefore, Ilya earns a commission of 80 for each policy sold during the week. We can then solve for the initial value. \begin{align*} I(n)&=80n+b \\ 760&=80(3)+b \text{ when } n=3, I(3)=760 \\ 760-80(3)&=b \\ 520 & =b \end{align*} The value of $$b$$ is the starting value for the function and represents Ilya’s income when $$n=0$$, or when no new policies are sold. We can interpret this as Ilya’s base salary for the week, which does not depend upon the number of policies sold. We can now write the final equation. $I(n)=80n+520 \nonumber$ Our final interpretation is that Ilya’s base salary is520 per week and he earns an additional \$80 commission for each policy sold.

##### Example $$\PageIndex{12}$$: Using Tabular Form to Write an Equation for a Linear Function

Table $$\PageIndex{1}$$ relates the number of rats in a population to time, in weeks. Use the table to write a linear equation.

 w, number of weeks P(w), number of rats 0 2 4 6 1000 1080 1160 1240

Solution

We can see from the table that the initial value for the number of rats is 1000, so $$b=1000$$.

Rather than solving for $$m$$, we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. This means that the rate of change is 80 rats per 2 weeks, which can be simplified to 40 rats per week.

$P(w)=40w+1000 \nonumber$

If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. For example, using $$(2,1080)$$ and $$(6,1240)$$

\begin{align*} m&=\dfrac{1240-1080}{6-2} \\ &=\dfrac{160}{4} \\ &= 40\end{align*}

Is the initial value always provided in a table of values like Table $$\PageIndex{1}$$?

No. Sometimes the initial value is provided in a table of values, but sometimes it is not. If you see an input of 0, then the initial value would be the corresponding output. If the initial value is not provided because there is no value of input on the table equal to 0, find the slope, substitute one coordinate pair and the slope into $$f(x)=mx+b$$, and solve for $$b$$.

$$\PageIndex{5}$$: A new plant food was introduced to a young tree to test its effect on the height of the tree. Table $$\PageIndex{2}$$ shows the height of the tree, in feet, $$x$$ months since the measurements began. Write a linear function, $$H(x)$$, where $$x$$ is the number of months since the start of the experiment.

 $$x$$ $$H(x)$$ 0 2 4 8 12 12.5 13.5 14.5 16.5 18.5

Solution

$$H(x)=0.5x+12.5$$

## Key Equations

• slope-intercept form of a line: $$f(x)=mx+b$$
• slope: $$m=\dfrac{\text{change in output (rise)}}{\text{change in input (run)}}=\dfrac{{\Delta}y}{{\Delta}x}=\dfrac{y_2-y_1}{x_2-x_1}$$
• point-slope form of a line: $$y−y_1 =m(x-x_1)$$

## Key Concepts

• The ordered pairs given by a linear function represent points on a line.
• Linear functions can be represented in words, function notation, tabular form, and graphical form.
• The rate of change of a linear function is also known as the slope.
• An equation in the slope-intercept form of a line includes the slope and the initial value of the function.
• The initial value, or y-intercept, is the output value when the input of a linear function is zero. It is the y-value of the point at which the line crosses the y-axis.
• An increasing linear function results in a graph that slants upward from left to right and has a positive slope.
• A decreasing linear function results in a graph that slants downward from left to right and has a negative slope.
• A constant linear function results in a graph that is a horizontal line.
• Analyzing the slope within the context of a problem indicates whether a linear function is increasing, decreasing, or constant.
• The slope of a linear function can be calculated by dividing the difference between y-values by the difference in corresponding x-values of any two points on the line.
• The slope and initial value can be determined given a graph or any two points on the line.
• One type of function notation is the slope-intercept form of an equation.
• The point-slope form is useful for finding a linear equation when given the slope of a line and one point.
• The point-slope form is also convenient for finding a linear equation when given two points through which a line passes.
• The equation for a linear function can be written if the slope $$m$$ and initial value $$b$$ are known.
• A linear function can be used to solve real-world problems.
• A linear function can be written from tabular form.

## Footnotes

1 www.chinahighlights.com/shang...glev-train.htm
2 www.cbsnews.com/8301-501465_1...ay-study-says/

## Glossary

decreasing linear function

a function with a negative slope: If $$f(x)=mx+b$$, then $$m<0$$.

increasing linear function

a function with a positive slope: If $$f(x)=mx+b$$, then $$m>0$$.

linear function

a function with a constant rate of change that is a polynomial of degree 1, and whose graph is a straight line

point-slope form

the equation for a line that represents a linear function of the form $$y−y_1=m(x−x_1) slope the ratio of the change in output values to the change in input values; a measure of the steepness of a line slope-intercept form the equation for a line that represents a linear function in the form \(f(x)=mx+b$$

y-intercept

the value of a function when the input value is zero; also known as initial value

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