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College Algebra
Tutorial 27: Graphing Lines


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Graph a straight line using the y-intercept and slope.
  2. Graph vertical and horizontal lines.




deskIntroduction



In this tutorial we will be looking at graphing lines.  There are several ways that you can approach graphing a line.  We will be looking specifically at using the y-intercept and slope to help us get our straight line.  I will provide a review on slope as well as the slope/intercept form of a line as we go through this lesson.  If you need more review on these concepts, feel free to go to Tutorial 25: Slope of a Line and Tutorial 26: Equations of Lines.  The concept of graphing vertical and horizontal lines will also be covered on this page.  Basically, we will plot the y-intercept point and then use the slope to find a second point and then connect the dots to get your graph.

 

 

desk Tutorial



 

Here is a little review on what the slope of a line tells about that line.  If you need more of a review on slope, feel free to go to Tutorial 25: Slope of a Line.


 

Slope

 
The slope of a line measures the steepness of the line.

Most of you are probably familiar with associating slope with "rise over run". 
 

Rise means how many units you move up or down from point to point.  On the graph that would be a change in the y values.

Run means how far left or right you move from point to point.  On the graph, that would mean a change of x values.


 

Here are some visuals to help you with this definition:
 
 

Positive slope:
positive slope

positive slope

Note that when a line has a positive slope it goes up left to right.


 
 
Negative slope:

negative slope

negative slope

Note that when a line has a negative slope it goes down left to right.


 

Zero slope:

zero slope

slope = 0 

Note that when a line is horizontal the slope is 0.


 

Undefined slope:

undefined slope

slope = undefined 

Note that when the line is vertical the slope is undefined.


 
 
Here is a little review on the slope/intercept form of the equation of a line.  If you need more of a review on slope, feel free to go to Tutorial 26: Equations of Lines.


 

Slope/Intercept Equation of a Line

y intercept


 
If your linear equation is written in this form, m represents the slope and b represents the y-intercept.

This form can be handy if you need to find the slope of a line given the equation.


 
 
 
Graphing a Line Using the y-intercept and Slope

 
Step 1: Put the equation in slope/intercept form (y intercept) and identify the slope and y-intercept.

 
When it is in this form it makes it easier to identify the slope and the y-intercept of the line. 

In this form, the slope is m, which is the coefficient in front of x and the y-intercept is the constant b.

 

Step 2: Plot the y-intercept on a 2-dimensional graph.

 
Recall that the y-intercept is where it crosses the y-axis.  So x's value will be 0.

If you need more review on intercepts, feel free to go to Tutorial 26: Equations of Lines.

 

Step 3: Use the slope to find a second point on the line.

 
You will start with the y-intercept and rise up or down and then run left or right depending on the sign of the slope. 

Use the concept of slope being rise over run (rise/run) to determine how to use the slope to find your second point.
 

What do you think is the positive direction for rise, up or down?   If you said up, pat yourself on the back.  That means down is the negative direction for rise.  A way to remember this is to think about the numbers are on the y-axis.  Going up above the origin are the positive values, and going down below the origin are the negative values. 

What do you think is the positive direction for run, right or left?   If you said right, pat yourself on the back.  That means left is the negative direction for run.  A way to remember this is to think about the numbers are on the x-axis.  Going to the right of  the origin are the positive values, and going to the left of the origin are the negative values. 
 

If the slope is positive, then the rise and the run need to either be BOTH positive or BOTH negative.  In other words, you will be going up and to the right OR down and to the left.  The reason both negative directions work is our slope is rise over run and if you have a negative over a negative, it simplifies to be a positive. 

If the slope is negative, then the rise and the run have to be opposites of each other, one has to be positive and one has to be negative.  In other words, you will be going up and to the left OR down and to the right.  If you make them both negative, then it would simplify to be a positive and you would have the wrong graph and you don't want to do that. 
 

Also keep in mind, if your slope is a non-zero integer like -5 or 10, that there is a denominator or run of your slope.  What is the denominator of a non-zero integer?  If you said 1, you are correct!!!

 

Step 4: Draw a line through the points found in steps 2 and 3.

 
All of the graphs in this tutorial will be straight lines.


 
 

notebook Example 1:  Give the slope and y-intercept of the line example 1a and then graph it.

 
Step 1: Put the equation in slope/intercept form (y intercept) and identify the slope and y-intercept.

 
This equation is already in slope/intercept form:

 
example 1b
*Slope/intercept form

 
Lining up the form with the equation we have been given, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x.  In our problem that would have to be 3. 

In this form, the y-intercept is b, which is the constant.  In our problem that would be 1

How did you do? 


 

 
Since the y-intercept is where the line crosses the y-axis, then x's value would have to be 0.  In step 1 we found our y-intercept value to be 1. 

Putting that together, the ordered pair for the y-intercept  would be (0, 1):


 
example 1c

 
 

 
In step 1 we found our slope to be 3.  What would the denominator of 3 be?  If you said 1, you are right on.  So we can think of 3 as 3/1.  That makes it easier to think of it as rise over run (rise/run). 

Since we have a positive slope, the rise and the run need to either be BOTH positive or BOTH negative.  So, we can either rise up 3 and run right 1 OR go down 3 and left 1. 

I chose to rise up 3 and run right 1, starting on the y-intercept::


 
example 1d

Note that if we would have gone down 3 and left 1 from our y-intercept, that we would have ended up at (-1, -2) which would have lined up with the other points.


 
 

 
example 1e

 
 
 
notebook Example 2:  Give the slope and y-intercept of the line example 2a and then graph it.

 
Step 1: Put the equation in slope/intercept form (y intercept) and identify the slope and y-intercept.

 
Solve for y to get it into the slope/intercept form:

 
example 2b

*Inverse of add. 3x is sub. 3x
 

*Inverse of mult. 2 is div. 2

*Slope/intercept form

 


 
Lining up the form with the equation we got, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x.  In our problem that would have to be -3/2. 

In this form, the y-intercept is b, which is the constant.  In our problem that would be 3

How did you do? 


 

 
Since the y-intercept is where the line crosses the y-axis, then x's value would have to be 0.  In step 1 we found our y-intercept value to be 3. 

Putting that together, the ordered pair for the y-intercept  would be (0, 3):


 
example 2c

 
 

 
In step 1 we found our slope to be -3/2. 

Since we have a negative slope the rise and the run have to be opposites of each other, one has to be positive and one has to be negative.  So, we can either go down 3 and run right 2 OR rise up 3 and run left 2. 

I chose to go down 3 and run right 2, starting on the y-intercept:


 
example 2d

Note that if we rose up 3 and ran left 2 from our y-intercept,  we would have ended up at (-2, 6) which would have lined up with the other points.


 
 

 
example 2e

 
 
 
notebook Example 3:  Give the slope and y-intercept of the line example 3a and then graph it.

 
Step 1: Put the equation in slope/intercept form (y intercept) and identify the slope and y-intercept.

 
This equation is already in slope/intercept form:

 
example 3b
*Slope/intercept form

 
Lining up the form with the equation we have been given, can you see what the slope and y-intercept are?

In this form, the slope is m, which is the number in front of x.  In our problem that would have to be 5/2. 

In this form, the y-intercept is b, which is the constant.  In our problem that would be 0

How did you do? 


 

 
Since the y-intercept is where the line crosses the y-axis, then x's value would have to be 0.  In step 1 we found our y-intercept value to be 0. 

Putting that together, the ordered pair for the y-intercept  would be (0, 0):


 
example 3c

 
 

 
In step 1 we found our slope to be 5/2. 

Since we have a positive slope the rise and the run need to either be BOTH positive or BOTH negative.  So, we can either rise up 5 and run right 2 OR go down 5 and left 2. 

I chose to rise up 5 and run right 2, starting on the y-intercept:


 
example 3d

Note that if we would have gone down 5 and left 2 from our y-intercept, we would have ended up at (-2, -5) which would have lined up with the other points.


 
 

 
example 3e


 
 
 

Vertical Lines

x = c


 
 
If you have an equation x = c, where c is a constant, and you are wanting to graph it on a two dimensional graph, this would be a vertical line with x-intercept of (c, 0). 

Even though you do not see a y in the equation, you can still graph it on a two dimensional graph.  Remember that the graph is the set of all solutions for a given equation.  If all the points are solutions, then any ordered pair that has an x value of c would be a solution.  As long as  x never changes value,  it is always c, then you have a solution.  In that case, you will end up with a vertical line. 
 
 

Below is an illustration of a vertical line x = c:

vertical slope

As mentioned above, the slope of a vertical line is undefined

Also, note that except for the vertical line x = 0, a vertical line does not go through the y-axis.  So that means a vertical line has no y-intercept, unless it is x = 0. 


 
 
 

Horizontal Lines

y = c


 
If you have an equation y = c, where c is a constant, and you are wanting to graph it on a two dimensional graph, this would be a horizontal line with y- intercept of (0, c). 

Even though you do not see an x in the equation, you can still graph it on a two dimensional graph.  Remember that the graph is the set of all solutions for a given equation.  If all the points are solutions, then any ordered pair that has a y value of c would be a solution.  As long as y never changes value,  it is always c, then you have a solution.  In that case, you will end up with a horizontal line. 

Below is an illustration of a horizontal line y = c:

horizontal line

As mentioned above, a horizontal line has a slope of 0

A horizontal line's y-intercept is whatever y is set equal to.


 
 
 
 
notebook Example 4:  Give the slope and y-intercept of the line example 4a and then graph it.

 
Note how we have an equation that has y set equal to a constant, where there is no x.  Since this fits the form of y = c described above, we can cut to the chase and draw our horizontal line:

 
example 4b

Since we have a horizontal line, what is our slope going to be?  If you said 0, you are so right!!!

What would the y-intercept be?  Give yourself a high five if you said 2.


 
 
notebook Example 5:  Give the slope and y-intercept of the line example 5a and then graph it.

 
Note how we are missing a y and even though it is not quite in the form x = c, that we can get it in that form. 

Lets first rewrite this in the form x = c and then go from there:


 
example 5b
*Written in the form x = c

 
Note how we have an equation that has x set equal to a constant, where there is no y.  Since this fits the form of x = c described above, we can jump to the chase and draw our vertical line:

 
example 5c

Since we have a vertical line, what is our slope going to be?  If you said undefined, you are so right!!!

What would the y-intercept be?  Give yourself a high five if you said there is no y-intercept.


  
 

desk Practice Problems



These are practice problems to help bring you to the next level.  It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it.  Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument.  In fact there is no such thing as too much practice.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that  problem.  At the link you will find the answer as well as any steps that went into finding that answer.

 

pencil Practice Problems 1a - 1d: Give the slope and y-intercept of the given line and then graph it.


 

 

 
 

 

desk Need Extra Help on these Topics?



The following are webpages that can assist you in the topics that were covered on this page:
 

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut15_slope.htm
This website helps you with the slope of a line.

http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut14_lineargr.htm#vertical
This website helps with vertical and horizontal lines.

http://www.purplemath.com/modules/slopgrph.htm
This webpage helps you with graphing a line using the y-intercept and the slope.


 


Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

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Last revised on Feb. 11, 2010 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.