Learning Objectives
Introduction
Tutorial
Most of you are probably familiar with associating slope with "rise
over run".
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:
Note that when a line has a positive slope it goes up left to right.
Note that when a line has a negative slope it goes down left to right.
slope = 0
Note that when a line is horizontal the slope is 0.
slope = undefined
Note that when the line is vertical the slope is undefined.
Slope/Intercept Equation of a Line
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
)
and identify the slope and y-intercept.In this form, the slope is m, which is the coefficient in front of x and the y-intercept is the constant b.
If you need more review on intercepts, feel free to go to Tutorial 26: Equations of Lines.
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!!!
Example
1: Give the slope and y-intercept
of the line 
and
then graph it.
)
and identify the slope and y-intercept.
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?
Putting that together, the ordered pair for the y-intercept
would be (0, 1):
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::
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
2: Give the slope and y-intercept
of the line 
and
then graph it.
)
and identify the slope and y-intercept.
*Inverse of mult. 2 is div. 2
*Slope/intercept form
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?
Putting that together, the ordered pair for the y-intercept
would be (0, 3):
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:
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
3: Give the slope and y-intercept
of the line 
and
then graph it.
)
and identify the slope and y-intercept.
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?
Putting that together, the ordered pair for the y-intercept
would be (0, 0):
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:
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.
x = c
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:
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.
y = 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:
As mentioned above, a horizontal line has a slope of 0.
A horizontal line's y-intercept is whatever y is set equal to.
Example
4: Give the slope and y-intercept
of the line 
and
then graph it.
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.
Example
5: Give the slope and y-intercept
of the line 
and
then graph it.
Lets first rewrite this in the form x = c and then go from there:
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.
Practice Problems
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.
Practice Problems 1a - 1d: Give the slope and y-intercept of the given line and then graph it.
Need Extra Help on these Topics?
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.
Last revised on Feb. 11, 2010 by Kim Seward.
All contents copyright (C) 2002 - 2010, WTAMU and Kim Seward. All rights reserved.
