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Linear Functions

Linear Functions

The simplest function we can graph on an -plane is a linear function. Even though they are simple, linear functions are still important! In AP Calculus, we study lines that are tangent to (or touching) curves, and when we zoom in enough on a curve, it looks and behaves like a line!

In this article, we discuss in detail what a linear function is, its characteristics, equation, formula, graph, table, and go through several examples.

  • Linear function definition
  • Linear function equation
  • Linear function formula
  • Linear function graph
  • Linear function table
  • Linear function examples
  • Linear functions - key takeaways

Linear Function Definition

What is a linear function?

A linear function is a polynomial function with a degree of 0 or 1. This means that each term in the function is either a constant or a constant multiplied by a single variable whose exponent is either 0 or 1.

When graphed, a linear function is a straight line in a coordinate plane.

By definition, a line is straight, so saying "straight line" is redundant. We use "straight line" often in this article, however, just saying "line" is sufficient.

Linear Function Characteristics

  • When we say that is a linear function of , we mean that the graph of the function is a straight line.

  • The slope of a linear function is also called the rate of change.

  • A linear function grows at a constant rate.

The image below shows:

  • the graph of the linear functionand
  • a table of sample values of that linear function.

Linear Functions the graph and table of sample values of a linear function StudySmarterThe graph and table of sample values of a linear function, StudySmarter Originals

Notice that when increases by 0.1, the value of increases by 0.3, meaning increases three times as fast as .

Therefore, the slope of the graph of , 3, can be interpreted as the rate of change of with respect to .

  • A linear function can be an increasing, decreasing, or horizontal line.

    • Increasing linear functions have a positive slope.

    • Decreasing linear functions have a negative slope.

    • Horizontal linear functions have a slope of zero.

  • The y-intercept of a linear function is the value of the function when the x-value is zero.

    • This is also known as the initial value in real-world applications.

Linear vs Nonlinear Functions

Linear functions are a special type of polynomial function. Any other function that does not form a straight line when graphed on a coordinate plane is called a nonlinear function.

Some examples of nonlinear functions are:

  • any polynomial function with a degree of 2 or higher, such as
    • quadratic functions
    • cubic functions
  • rational functions
  • exponential and logarithmic functions

When we think of a linear function in algebraic terms, two things come to mind:

  • The equation and

  • The formulas

Linear Function Equation

A linear function is an algebraic function, and the parent linear function is:

Which is a line that passes through the origin.

In general, a linear function is of the form:

Where and are constants.

In this equation,

  • is the slope of the line
  • is the y-intercept of the line
  • is the independent variable
  • or is the dependent variable

Linear Function Formula

There are several formulas that represent linear functions. All of them can be used to find the equation of any line (except vertical lines), and which one we use depends on the available information.

Since vertical lines have an undefined slope (and fail the vertical line test), they are not functions!

Standard Form

The standard form of a linear function is:

Where are constants.

Slope-intercept Form

The slope-intercept form of a linear function is:

Where:

  • is a point on the line.

  • is the slope of the line.

    • Remember: slope can be defined as , where and are any two points on the line.

Point-slope Form

The point-slope form of a linear function is:

Where:

  • is a point on the line.

  • is any fixed point on the line.

Intercept Form

The intercept form of a linear function is:

Where:

  • is a point on the line.

  • and are the x-intercept and the y-intercept, respectively.

Linear Function Graph

The graph of a linear function is pretty simple: just a straight line on the coordinate plane. In the image below, the linear functions are represented in slope-intercept form. (the number that the independent variable, , is multiplied by), determines the slope (or gradient) of that line, and determines where the line crosses the y-axis (known as the y-intercept).

Linear Functions graph of two lines on the same coordinate plane StudySmarterThe graphs of two linear functions, StudySmarter Originals

Graphing a Linear Function

What information do we need to graph a linear function? Well, based on the formulas above, we need either:

  • two points on the line, or

  • a point on the line and its slope.

Using Two Points

To graph a linear function using two points, we need to either be given two points to use, or we need to plug in values for the independent variable and solve for the dependent variable to find two points.

  • If we are given two points, graphing the linear function is just plotting the two points and connecting them with a straight line.

  • If, however, we are given a formula for a linear equation and asked to graph it, there are more steps to follow.

Graph the function:

Solution:

  1. Find two points on the line by choosing two values for .
    • Let's assume values of and .
  2. Substitute our chosen values of into the function and solve for their corresponding y-values.
    • So, our two points are: and .
  3. Plot the points on a coordinate plate, and connect them together with a straight line.
    • Be sure to extend the line past the two points, as a line is never-ending!
    • So, the graph looks like:
    • Linear Functions graphing a line using two points StudySmarter
      The graph of a line using two points, StudySmarter Originals

Using Slope and y-intercept

To graph a linear function using its slope and y-intercept, we plot the y-intercept on a coordinate plane, and use the slope to find a second point to plot.

Graph the function:

Solution:

  1. Plot the y-intercept, which is of the form: .
    • The y-intercept for this linear function is:
  2. Write the slope as the fraction (if it isn't one already!) and identify the "rise" and the "run".
    • For this linear function, the slope is .
      • So, and .
  3. Starting at the y-intercept, move vertically by the "rise" and then move horizontally by the "run".
    • Note that: if the rise is positive, we move up, and if the rise is negative, we move down.
    • And note that: if the run is positive, we move right, and if the run is negative, we move left.
    • For this linear function,
      • We "rise" up by 1 unit.
      • We "run" right by 2 units.
  4. Connect the points with a straight line, and extend it past both points.
    • So, the graph looks like:
    • Linear Functions a graph of a line using its y-intercept and slope StudySmarter
      Using the slope and y-intercept to graph a line, StudySmarter Originals

Domain and Range of a Linear Function

So, why do we extend the graph of a linear function past the points we use to plot it? We do that because the domain and range of a linear function are both the set of all real numbers!

Domain

Any linear function can take any real value of as an input, and give a real value of as an output. This can be confirmed by looking at the graph of a linear function. As we move along the function, for every value of , there is only one corresponding value of .

Therefore, as long as the problem doesn't give us a limited domain, the domain of a linear function is:

Range

Also, the outputs of a linear function can range from negative to positive infinity, meaning that the range is also the set of all real numbers. This can also be confirmed by looking at the graph of a linear function. As we move along the function, for every value of , there is only one corresponding value of .

Therefore, as long as the problem doesn't give us a limited range, and , the range of a linear function is:

When the slope of a linear function is 0, it is a horizontal line. In this case, the domain is still the set of all real numbers, but the range is just b.

Linear Function Table

Linear functions can also be represented by a table of data that contains x- and y-value pairs. To determine if a given table of these pairs is a linear function, we follow three steps:

  1. Calculate the differences in the x-values.

  2. Calculate the differences in the y-values.

  3. Compare the ratio for each pair.

    • If this ratio is constant, the table represents a linear function.

We can also check if a table of x- and y-values represents a linear function by determining if the rate of change of with respect to (also known as the slope) remains constant.

Typically, a table representing a linear function looks something like this:

x-valuey-value
14
25
36
47

Identifying a Linear Function

To determine if a function is a linear function depends on how the function is presented.

  • If a function is presented algebraically:

    • then it is a linear function if the formula looks like: .

  • If a function is presented graphically:

    • then it is a linear function if the graph is a straight line.

  • If a function is presented using a table:

    • then it is a linear function if the ratio of the difference in y-values to the difference in x-values is always constant. Let's see an example of this

Determine if the given table represents a linear function.

x-valuey-value
315
523
731
1147
1355

Solution:

To determine if the values given in the table represent a linear function, we need to follow these steps:

  1. Calculate the differences in x-values and y-values.
  2. Calculate the ratios of difference in x over difference in y.
  3. Verify whether the ratio is the same for all X,Y pairs.
    • If the ratio is always the same, the function is linear!

Let's apply these steps to the given table:

Linear Functions determining if a table of values represents a linear funcion StudySmarterDetermining if a table of values represents a linear function, StudySmarter Originals

Since every number in the green box in the image above are the same, the given table represents a linear function.

Special Types of Linear Functions

There are a couple of special types of linear functions that we will likely deal with in calculus. These are:

  • Linear functions represented as piecewise functions and

  • Inverse linear function pairs.

Piecewise Linear Functions

In our study of calculus, we will have to deal with linear functions that may not be uniformly defined throughout their domains. It could be that they are defined in two or more ways as their domains are split into two or more parts.

In these cases, these are called piecewise linear functions.

Graph the following piecewise linear function:

The symbol ∈ above means "is an element of".

Solution:

This linear function has two finite domains:

  • and

Outside of these intervals, the linear function does not exist. So, when we graph these lines, we will actually just graph the line segments defined by the endpoints of the domains.

  1. Determine the endpoints of each line segment.
    • For the endpoints are when and .
    • Notice in the domain of x+2 that there is a parenthesis instead of a bracket around the 1. This means that 1 is not included in the domain of x+2! So, there is a "hole" in the function there.

    • For the endpoints are when and .
  2. Calculate the corresponding y-values at each endpoint.
    • On the domain :
      • x-valuey-value
        -2
        1
    • On the domain :
      • x-valuey-value
        1
        2
  3. Plot the points on a coordinate plane, and join the segments with a straight line.
    • Linear Functions graph of piecewise linear function StudySmarter
      The graph of a piecewise linear function, StudySmarter Originals

Inverse Linear Functions

Likewise, we will also deal with inverse linear functions, which are one of the types of Inverse Functions. To briefly explain, if a linear function is represented by:

Then its inverse is represented by:

such that

The superscript, -1, is not a power. It means "the inverse of", not "f to the power of -1".

Find the inverse of the function:

Solution:

  1. Replace with .
  2. Replace with , and with .
  3. Solve this equation for .
  4. Replace with .

If we graph both and on the same coordinate plane, we will notice that they are symmetric with respect to the line . This is a characteristic of Inverse Functions.

Linear Functions graph of inverse linear functions and their line of symmetry StudySmarterThe graph of an inverse linear function pair and their line of symmetry, StudySmarter Originals

Linear Function Examples

Real-World Applications of Linear Functions

There are several uses in the real world for linear functions. To name a few, there are:

  • Distance and rate problems in physics

  • Calculating dimensions

  • Determining prices of things (think taxes, fees, tips, etc. that are added to the price of things)

Say you enjoy playing video games.

You subscribe to a gaming service that charges a monthly fee of $5.75 plus an additional fee for each game you download of $0.35.

We can write your actual monthly fee using the linear function:

Where is the number of games you download in a month.

Linear Functions: Solved Example Problems

Write the given function as ordered pairs.

Solution:

The ordered pairs are: and .

Find the slope of the line for the following.

Solution:

  1. Write the given function as ordered pairs.
  2. Calculate the slope using the formula: , where correspond to respectively.
    • , so the slope of the function is 1.

Find the equation of the linear function given by the two points:

Solution:

  1. Using the slope formula, calculate the slope of the linear function.
  2. Using the values given by the two points, and the slope we just calculated, we can write the equation of the linear function using point-slope form.
    • - point-slope form of a line.
    • - substitute in values for .
    • - distribute the negative sign.
    • - distribute the 4.
    • - simplify.
    • is the equation of the line.

The relationship between Fahrenheit and Celsius is linear. The table below shows a few of their equivalent values. Find the linear function representing the given data in the table.

Celsius (°C)Fahrenheit (°F)
541
1050
1559
2068

Solution:

  1. To start, we can pick any two pairs of equivalent values from the table. These are the points on the line.
    • Let's choose and .
  2. Calculate the slope of the line between the two chosen points.
    • , so the slope is 9/5.
  3. Write the equation of the line using point-slope form.
    • - point-slope form of a line.
    • - substitute in values for .
    • - distribute the fraction and cancel terms.
    • - simplify.
  4. Note that based on the table,
    • We can replace , the independent variable, with , for Celsius, and
    • We can replace , the dependent variable, with , for Fahrenheit.
    • So we have:
      • is the linear relationship between Celsius and Fahrenheit.

Let's say that the cost of renting a car can be represented by the linear function:

Where is the number of days the car is rented.

What is the cost to rent the car for 10 days?

Solution:

  1. Substitute into the given function.
    • - substitute.
    • - simplify.

So, the cost of renting the car for 10 days is $320.

To add onto the last example. Let's say we know how much someone paid to rent a car, using the same linear function.

If Jake paid $470 to rent a car, how many days did he rent it?

Solution:

We know that , where is the number of days the car is rented. So, in this case, we replace with 470 and solve for .

  1. - substitute known values.
  2. - combine like terms.
  3. - divide by 30 and simplify.
  4. So, Jake rented the car for 15 days.

Determine if the function is a linear function.

Solution:

We need to isolate the dependent variable to help us visualize the function. Then, we can verify whether it is linear by graphing it.

  1. - move all terms except the dependent variable to one side of the equation.
  2. - divide by -2 to simplify.
    • Now, we can see that the independent variable, , has a power of 1. This tells us that this is a linear function.
  3. We can verify our findings by drawing the graph:
    • Linear Functions graph of a line StudySmarterThe graph of a line, StudySmarter Originals

Determine whether the function is a linear function.

Solution:

  1. Rearrange and simplify the function to get a better visualization.
    • - distribute the .
    • - move all terms except the dependent variable to one side.
    • - divide by 2 to simplify.
  2. Now, we can see that since the independent variable has a power of 2, this is not a linear function.
  3. We can verify that the function is nonlinear by graphing it:
    • Linear Functions graph of a nonlinear function StudySmarterThe graph of a nonlinear function, StudySmarter Originals

Linear Functions - Key takeaways

  • A linear function is a function whose equation is: and its graph is a straight line.
    • A function of any other form is a nonlinear function.
  • There are forms the linear function formula can take:
    • Standard form:
    • Slope-intercept form:
    • Point-slope form:
    • Intercept form:
  • If the slope of a linear function is 0, it is a horizontal line, which is known as a constant function.
  • A vertical line is not a linear function because it fails the vertical line test.
  • The domain and range of a linear function is the set of all real numbers.
    • But the range of a constant function is just , the y-intercept.
  • A linear function can be represented using a table of values.
  • Piecewise linear functions are defined in two or more ways as their domains are split into two or more parts.
  • Inverse linear function pairs are symmetric with respect to the line .
    • A constant function has no inverse because it is not a one-to-one function.

Frequently Asked Questions about Linear Functions

A linear function is an algebraic equation in which each term is either:

  • a constant (just a number) or
  • the product of a constant and a single variable that has no exponent (i.e. that is to the power of 1)


The graph of a linear function is a straight line.


For example, the function: y = x is a linear function.

  • Using its graph, you can write a linear function by finding the slope and y-intercept.
  • Given a point and a slope, you can write a linear function by:
    • plugging the values from the point and slope into the slope-intercept form of the equation of a line: y=mx+b
    • solving for b
    • then writing the equation
  • Given two points, you can write a linear function by:
    • calculating the slope between the two points
    • using either point to calculate b
    • then writing the equation

To determine if a function is a linear function, you need to either:

  • verify that the function is a first-degree polynomial (the independent variable must have an exponent of 1)
  • look at the graph of the function and verify that it is a straight line
  • if given a table, calculate the slope between each point and verify that the slope is the same

Considering the following table:

x: 0, 1, 2, 3

y: 3, 4, 5, 6


From this table, we can observe that the rate of change between x and y is 3. This can be written as the linear function: y = x + 3.

Final Linear Functions Quiz

Question

What is a linear function?

Show answer

Answer

A linear function is a polynomial function with a degree of 0 or 1. This means that each term in the function is either a constant or a constant multiplied by a single variable whose exponent is either 0 or 1.


When graphed, a linear function is a straight line in a coordinate plane.

Show question

Question

The slope of a linear function is also called...

Show answer

Answer

the rate of change.

Show question

Question

A linear function can be...

Show answer

Answer

an increasing, decreasing, or horizontal line.

Show question

Question

Increasing linear functions have...

Show answer

Answer

a positive slope.

Show question

Question

Decreasing linear functions have...

Show answer

Answer

a negative slope.

Show question

Question

Horizontal linear functions have...

Show answer

Answer

zero slope.

Show question

Question

The y-intercept of a linear function is...

Show answer

Answer

the value of the function when the x-value is zero.

  • This is also known as the initial value.

Show question

Question

Some examples of nonlinear functions are:

Show answer

Answer

any polynomial function with a degree of 2 or higher, such as

  • quadratic functions
  • cubic functions

Show question

Question

Are vertical lines linear functions?

Show answer

Answer

No!

Since vertical lines have an undefined slope and fail the vertical line test they are not functions!

Show question

Question

What are the 4 formulas that can represent a linear function?

Show answer

Answer

  1. Standard form
  2. Slope-intercept form
  3. Point-slope form
  4. Intercept form

Show question

Question

What two ways can we use to graph a linear function?

Show answer

Answer

Using two points.

Show question

Question

In our study of calculus, we will have to deal with linear functions that may not be uniformly defined throughout their domains. It could be that they are defined in two or more ways as their domains are split into two or more parts.


In these cases, these are called...

Show answer

Answer

piecewise linear functions.

Show question

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