Linear Functions

Created by: Greg O'Hearn, Jeremy Sheaffer, and Shannon PenixRevised and Expanded by: Nick Schweitzer, Brian Hurner, Marco Iaderosa, Marty BrudzinksiHeading Images Created by: Nick Schweitzer

So...What is a Linear Function?

According to the Merriam-Webster dictionary, a linear function is "a mathematical function in which the variables only appear in the first degree, are multiplied by constants, and are combined only by addition and subtraction." We will explore the various equation forms of linear functions and their applications below.

Slope-Intercept Form:

A linear function is in slope-intercept form when it is of the form y = mx + b such that m is the slope, b is the y-coordinate of the y-intercept, and m and b are both real numbers. The line represents the set of all ordered pairs, (x, y), with real number entries x and y such that the equation of the line is satisfied. For example, the equation y = 3x + 1 is a linear function in slope-intercept form, since 3 and 1 are real numbers and it is in the form y = mx + b. The ordered pair (2,7) is a solution to the equation, since 7 = 3(2) + 1 and 2 and 7 are real numbers. Some of the key features of a linear function in slope-intercept form are its slope and its possible intercept(s).

We will first discuss slope, which can be defined as the ratio of the vertical change to horizontal change of a line. Some other ways to describe slope are
  • the direction or steepness of the line,
  • the rate of change of a line,
  • the increase or decrease in y for a unit change in x,
  • the amount the output variable is changing for every unit change in the input variable, and,
  • algebraically,

wiki slop image.png
Image created by Nick Schweitzer

As stated before, slope is real number. In fact, it is a constant value that does not change its value. This can be seen graphically through what could be called the "staircase effect" of slope with the image below. Each "step" of the staircase is the same, thus demonstrating that linear functions have a constant slope.

slope staircase.png
Image created by Nick Schweitzer
A few properties of slope are

    • if m is positive, then the line is increasing and it will go up and to the right graphically,
    • if m is negative, then the line is decreasing and it will go down and the the right graphically,
    • if m is zero, then the line is a horizontal line that extends infinitely to the left and right at the value of b, and
    • if the line is a vertical line, then the slope is undefined and graphically it extends infinitely up and down for the given x value.

The next key feature we will discuss are the possible intercept(s) of a linear function. There are two types of intercepts: vertical intercepts, or y-intercepts, and horizontal intercepts, or x-intercepts. By definition, the y-intercept occurs where the graph of the line intersects the y-axis, which occurs when x is zero. Thus, the y-intercept is (0, b) for a linear function in slope-intercept form because y = m(0) + b = b. It is important to note that vertical lines do not have y-intercepts except for x = 0 because the vertical lines does not have an x-coordinate of 0 anywhere on the graph. Similarly, by definition, the x-intercept occurs where the graph of the line intersects the x-axis, which occurs when y is zero. Thus, the y-intercept is (-b/m, 0) for a linear function in slope-intercept form because 0 = mx + b, and through solving for x using standard algebra, we find x = -b/m. It is also important to note that horizontal lines do not have x-intercepts except for y = 0 because the horizontal line does not have a y-coordinate of 0 anywhere on the graph.

Example Student Problem:

Given the two points (9,3) and (7,2), find the equation of the line that contains these ordered pairs in slope intercept form, (y=mx+b).
Step 1: Find the slope the equation, m.
To begin, we will substitute the entries of the ordered pairs into the slope formula, {m=(y_2 - y_1)/(x_2 - x_1)}.
We will call (9, 3) the first ordered pair and (7, 2) the second ordered pair. Therefore,
m=(3 - 2)/(9 - 7)= 1/2.
Thus, the equation of the line containing the two points, so far, is y = (1/2)x + b.

Step 2: Find the y-intercept of the equation to locate the value of b.
Now, we will substitute either one of ordered pairs into y=(1/2)x+b and solve for b. We will use (9, 3). Thus, using standard algebra, we find
Since we know b= -1.5, the equation of our line in slope-intercept form is y = (1/2)x - 1.5.

Here are some teacher resources related to slope-intercept form.

Parallel and Perpendicular Lines

We will now discuss parallel and perpendicular lines, since they are most easily found algebraically from slope-intercept form. First, we will discuss parallel lines, which can be defined as two different lines whose slopes are equal. Algebraically, this means that the value of m in the two equations are the same. Graphically, as seen in the image below, the lines are going in the same direction and will never intersect.

wiki parallel creation.png
Image created by Nick Schweitzer

Next, we will discuss perpendicular lines, which can be defined as two different lines whose slope are opposite reciprocals of each other. Algebraically, this means that

Image created by Nick Schweitzer

and geometrically, the two lines intersect to form a 90 degree angle, as shown in the image below.

wiki perpendicular creation.png
Image created by Nick Schweitzer

Here are some resources,including exploration activities and lesson plans, that involve parallel and perpendicular lines.

This activity was created by Nick Schweitzer and provides a worksheet where students find the equation of lines in slope-intercept form given two points, the slope and a point, a table of points, or a graph, determining if two lines are parallel or perpendicular, and creating the equation of a line that is parallel or perpendicular to a given line and is through a given point. This resource touches on the Common Core standards HSF.BF.A.1 and HSG.GPE.B.5. Any questions about or complications with the file can be directed to Nick at

Constant Linear Functions

Definition: A constant linear function is a function that does not contain a variable.

Key features: A constant linear function can be a horizontal line that takes the form y = m where m is a real number. It also can be a vertical line that takes the form x = n where n is a real number. Unlike other linear functions in this section, a constant linear function does not have a slope. Therefore it appears as straight vertical or horizontal line on a graph.

Graph (examples) Looking at a few examples we can see how some functions are graphed


As we can see this is a constant linear function because it is a straight horizontal line. So it takes the form y = m. The line is 2 units above the x-axis and therefore the equation is y = 2.



This a vertical line so it takes the form x = n. It is 3 units to the right of the y-axis so the equation is x = 3.


Constant linear functions only have one use in algebra. Since they are only constant terms, they can be plugged into more complicated functions.

Example 1:
y = x^3-4x+7
x = 2

The x function can be plugged into the y function.

y = (2)^3 - 4(2) + 7
y = 7

Example 2:
x = y^2 + 2x + 5
y = 3

The y function can be plugged into the x function.

x = (3)^2 + 2(3) + 5
y = 20

The link above has a bunch of different worksheets that can be used to in a classroom to help teach linear functions.

This activity is to give out a graph to the students. Then play one of the videos on the site. They are videos of different real life actions that are relevant to linear functions.

__CCSS.MATH.CONTENT.HSF.IF.B.4__. This is the common core standard it covers.

Systems of Linear Equations

A "system" of linear equations is a set or collection of linear equations that you deal with all together at once. To solve a system of linear equations, you must solve for multiple variables. One rule for solving a system of linear equations is that the number of unknown variables must equal the number of equations. Thus if need to solve for 4 variables, there must be 4 equations. There are two ways to solve a system of linear equations.

The first is by substitution.

2x - 3y = 5
2x - 7y = 7

solve for x in the first equation

2x - 3y = 5
2x = 5 + 3y
x = (5 + 3y)/2

substitute into the second equation and solve for y

2[(5+3y)/2] - 7y = 7
(10+6y)/2 - 7y = 7
10 + 6y - 14y = 14
10 - 8y =14
-8y = 4
y = -(½)

Now plug y into either equation and solve for x

2x - 3(-1/2) = 5
2x = 3.5
x = 1.75

The second way is to solve using elimination

2x - 3y = 5
2x - 7y = 7

Subtract the two equations(notice the X’s cancel)

4y = -2
y = -(½)

then substitute into either equation to solve for x
2x - 3(-½) = 5
2x = 3.5
x = 1.75

There are many more examples on this link __


There are situations all around us that use linear equations. Anything that has a constant rate of change will follow the form of a linear function. This site has multiple examples of where we can find real life applications of linear functions: __

While the site above has some simple examples with minimal explanation, this site has more complicated problems with much more detail. There are more specific examples and much more information: __

Point-Slope Form

Definition: The equation of a straight line in the form y − y1 = m(x − x1) where m is the slope of the line and (x1, y1) are the coordinates of a given point on the line.

Key Features: Point-Slope form makes it easy to find the line's equation when you only know the slope and a single point on the line. It is also the quickest method for finding the equation of line given two points.

Graphically: Here are some examples of graphs in which point-slope form can be used.

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This graph is a great example because it shows two sets of points connected by a single straight line. This information is enough in order to construct a function that describes the line through these two points (which will be covered in the algebra section)

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This graph is another great example of using given information to construct a function to represent the straight line connecting the two blue points. Here, the slope is given and one point is clearly given (though the other can easily be seen). That information is enough to construct our desired function.

For this problem, they give you a point (x1, y1) and a slope m, and have we plug it into this formula:
  • y – y1 = m(x – x1)
They are just intended to indicate the point they give us. We have the generic "x" and generic "y" that are always in our equation, and then we have the specific x and y from the point they gave us; the specific x and y are what is subscripted in the formula. Here's how we use the point-slope formula:

Find the equation of the straight line that has slope m = 4 and passes through the point (–1, –6).
  • We want to put this information into the point slope formula.
  • They've given us m = 4, x1

    –1, and y1

    –6. We plug these values into the point-slope form.
    • y – y1 = m(x – x1)
    • y – (–6) = (4)(x – (–1))
    • y + 6 = 4(x + 1)
Done! We can take this a step further by converting this into standard form by solving for “y”
  • y + 6 = 4x + 4
  • y = 4x + 4 – 6
  • y = 4x – 2
Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

Let’s do another!!
We can find the straight-line equation using the point-slope form if they just give us a couple points:

Find the equation of the line that passes through the points (–2, 4) and (1, 2).
  • Given two points, we can always find the slope:
external image pgKE5FeiMwn7Zf-3WC1Rb1oF7OmFeq37aFDf_zse_XkWLrt2HPO4KsBrMz9Oafs6xaE7rjY3VwmKWiaGXiiKNiDnRck7ojsP8M4caEMaCD0fB2EM9uIpvghLJK0uEG_4f8W6tA
    • slope m = -2/3
      slope m = -2/3
  • Then we can use either point as my (x1, y1), along with this slope we have just calculated, and plug in to the point-slope form. Using (–2, 4) as the (x1, y1), we get:
  • y – y1 = m(x – x1)
  • y – (4) = ( – 2/3 )(x – (–2))
  • y – 4 = ( – 2/3 )(x + 2)
  • y – 4 = ( – 2/3 ) x – 4/3
  • y = ( – 2/3 ) x – 4/3 + 4
  • y = ( – 2/3 ) x – 4/3 + 12/3
  • y = ( – 2/3 ) x + 8/3



Standard Form
The Standard Form equation of a line has the following formula: Ax + By = C where A≠0 and B≠0

Key Features:
A shouldn't be negative, A and B shouldn't both be zero, and A, B and C should be integers.
The standard form is not nearly as useful as the slope intercept or point slope forms. These forms focus on key components of a linear equation where the standard form is the most general way to display a linear equation.
Finding the intercepts for a linear equation is easiest when the equation is in standard form.
The Y-Intercept of a line is the point where a line's graph intersects (crosses) the Y-axis.
•A y-intercept of 3 means that a line's graph intersects the Y-axis at the point (0,3).
•A y-intercept of -4 means that the graph of a line crosses the Y-axis at the point (0,-4).
An x-intercept of 5 , means that the line cross the x-axis at 5, which is the point ( 5,0)
• An x-intercept of -2 , means that the line cross the x-axis at -2, which is the point ( -2,0)
• An x-intercept of 3 , means that the line cross the x-axis at 3, which is the point ( 3,0)

1.Find the intercepts of the following equation: 3x + 2y = 6
How to find the x intercept
Set y = 0 3x + 2(0) = 6
Solve for x
x = 2
How to find the y -intercept:
Set x = 0 3(0) + 2y = 6
Solve for y
y = 3.

2.The standard form of this linear equation is 3x – 2y = 6. This is the algebraic example of the graph below.

3.The algebraic form of this linear equation is given on the left of the graph below. This also gives an example of how to find the x and y intercepts of the standard form of linear equations.


Common Core Standards Involving Linear Functions:

Disclaimer: Not all of the standards have related content or resources on this page. For example, comparing linear functions to other function families or inverses of linear functions. Hopefully, with more revisions to the page, they will arrive shortly. Here are all CCSS math standards. A list of all Common Core standards pertaining to linear function can be found here.


Clark, Mark, and Cynthia Afinson. Intermediate Algebra: Connecting Concepts Through Applications. Belmont, CA: Brooks/Cole, 2012. 22-70. Print.