Intermediate Algebra
Tutorial 14:
Graphing Linear Equations
 Learning Objectives
|
After completing this tutorial, you should be able to:
- Find the x- and y-intercepts
of a linear function.
- Graph a linear function using the x-
and y-intercepts.
- Graph vertical and horizontal lines.
|
Introduction
|
| In Tutorial 12:
Graphing Equations,
we went over graphing in general, learning the basics of how to graph
ANY
equation by plotting points. In this tutorial we will be looking
specifically at graphing lines. The concept of using intercepts
to
help graph will be introduced on this page, as well as vertical and
horizontal
lines. Actually, the process of graphing by plotting points
and
graphing by using intercepts are essentially the same. Intercepts
are just special types of solutions, but solutions none the less.
So once we find them, we plot them just the same as any other ordered
pair
that is a solution. Once we plot them, we draw our graph in the
same
fashion as when we had non-intercept points. So, basically, when
you graph, you plot solutions (whether they are intercept points or
not)
and connect the dots to get your graph. See
graphing can be fun, it combines math and art together. |
Tutorial
|
| The x-intercept is
where the graph crosses
the x axis.
The word 'intercept' looks like the word
'intersect'. Think
of it as where the graph intersects the x-axis.
With that in mind, what
value is y
always going to be on the x-intercept?
No matter where you are on the x-axis, y’s
value is 0, that is a constant. We will use that
bit of information to help us find the x-intercept
when given an equation.
|
| If the x-intercept is
where the graph crosses
the x-axis where do you think the
graph
crosses for the y-intercept?
If you
said the y-axis, you are
absolutely
right.
This time it is x’s
value that is 0. Any where you would cross the y-axis, x’s
value is always 0. We will use this tidbit to help us find the y-intercept
when given an equation.
Below is an illustration of a graph of a linear function
which highlights
the x and y intercepts:
In the above illustration, the x-intercept
is the point (2, 0) and the y-intercept
is the point (0, 3).
Keep in mind that the x-
and y- intercepts are two separate
points. There
is only one point that can be both an x-
and y-
intercept at the same time, do you know what point that is?
If you said the origin (0, 0), give yourself a
pat on the back.
|
Sketching a Graph
Using Intercepts
|
| Step 1: Find the x-
and y- intercepts. |
| You find the x-intercept
by plugging in
0 for y and solving for x.
You find the y-intercept
by plugging in
0 for x and solving for y.
|
| Step 2: Find at least one more
point. |
| Step 3: Plot the intercepts
and point(s) found
in steps 1 and 2. |
| Remember that intercepts are points on the graph,
too. They are
plotted just like any other point. |
| The graph of a linear function is a straight line. |
 Example
1: Graph each linear function by finding the x-
and y- intercepts. y=
5 - 3x |
Let’s first find the x-intercept.
What value are we going to use for y?
You are correct if you said y = 0. |
 |
*Find x-int.
by
replacing y with 0
*Inverse of add 5 is sub. 5
*Inverse of mult. by -3 is div.
by -3
|
The x-intercept is
(5/3, 0).
Next, we will find the y-
intercept.
What value are we going to plug in for x?
If you said x = 0 you are right.
|
 |
*Find y-int.
by
replacing x with 0
|
| The y-intercept is
(0, 5) |
| We can plug in any x value
we want as long
as we get the right corresponding y
value and
the function exists there.
Let’s put in an easy number x
=
1:
|
 |
*Replace x
with
1
|
| So the ordered pair (1, 2) is another solution
to our function.
Note that we could have plugged in any value for x: 5,
10, -25, ...,
but it is best to keep it as simple as possible.
The solutions that we found are:
|
x
|
y
|
(x, y)
|
|
5/3
|
0
|
(5/3, 0)
|
|
0
|
5
|
(0, 5)
|
|
1
|
2
|
(1, 2)
|
|
| Example
2: Graph each linear function by finding the x-
and y- intercepts. -3x
= y |
Let’s first find the x-intercept.
What value are we going to use for y?
You are correct if you said y = 0. |
 |
*Find x-int.
by
replacing y with 0
*Inverse of mult. by -3 is div. by
-3 |
The x-intercept is
(0, 0).
Next, we will find the y-
intercept.
What value are we going to plug in for x?
If you said x = 0, you are right.
|
 |
*Find y-int.
by
replacing x with 0 |
| The y-intercept is
(0, 0)
Hey, look at that, we ended up with the exact same point
for both our x-
and y-intercepts. As mentioned
above,
there is only one point that can be both an x-
and y- intercept at the same time, the
origin
(0, 0).
|
| Since we really have found only one point this time,
we better find
two additional solutions so we have a total of three points.
We can plug in any x value
we want as long
as we get the right corresponding y value
and
the function exists there.
Let’s put in an easy number x
= 1:
|
 |
*Replace x with
1 |
So the ordered pair (1, -3) is another solution
to our function.
Let’s put in another easy number x = -1:
|
 |
*Replace x
with
-1 |
| So the ordered pair (-1, 3) is another solution
to our function.
Note that we could have plugged in any value for x: 5,
10, -25, ...,
but it is best to keep it as simple as possible.
The solutions that we found are:
|
x
|
y
|
(x, y)
|
|
0
|
0
|
(0, 0)
|
|
1
|
-3
|
(1, -3)
|
|
-1
|
3
|
(-1, 3)
|
|
| 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:
|
| 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 an 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:
|
| Example
3: Graph the linear equation y
= 4. |
It looks like it fits the form y
= c.
With that in mind, what kind of line are we going to end up with?
Horizontal.
Note how the directions did not specify that we had
to use intercepts
to do our graph. Any time you take a math test or do
homework,
make sure that you follow directions carefully. If it specifies a
certain way to do a problem, then you need to follow that plan (like in
the above examples 1 and 2). If it does not specify, like in this
example, then you can use what ever “legitimate” way works to get the
job
done.
|
| Since this is a special type of line, I thought I would
talk about
steps 1 and 2 together.
It doesn’t matter what x
is, y
is always 4. So for our solutions we just need three ordered
pairs
such that y = 4.
Note that the y-intercept
(where x =
0) is at (0, 4).
Do we have a x-intercept? The answer is no.
Since y
has to be 4, then it can never equal 0, which is the criteria of an x-intercept.
Also, think about it, if we have a horizontal line that crosses the y-axis
at 4, it will never ever cross the x-axis.
So, some points that we can use are (0, 4), (1, 4)
and (2, 4).
These are all ordered pairs that fit the criteria of y
having to be 4.
Of course, we could have used other solutions, there are
an infinite
number of them.
The solutions that we found are:
|
x
|
y
|
(x, y)
|
|
0
|
4
|
(0, 4)
|
|
1
|
4
|
(1, 4)
|
2
|
4
|
(2, 4)
|
|
| Example
4: Graph the linear equation x
+ 3 = 0. |
Note how if we subtract 3 from both sides, we can write
this as x
= -3, which means it can be written in the form x
= c.
So, what type of line are we going to end up with?
Vertical. |
| Since this is a special type of line, I thought I would
talk about
steps 1 and 2 together.
It does not matter what y
is, as long as x
is -3.
Note that the x-intercept
is at (-3,
0).
Do we have a y-intercept?
The answer is no. Since x can
never
equal 0, then there will be no y-intercept for this equation.
Some points that would be solutions are (-3, 0), (-3,
1), and (-3,
2).
Again, I could have picked an infinite number of
solutions.
The solutions that we found are:
|
x
|
y
|
(x, y)
|
|
-3
|
0
|
(-3, 0)
|
|
-3
|
1
|
(-3, 1)
|
|
-3
|
2
|
(-3, 2 )
|
|
Practice Problems
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| 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.
|
 Practice
Problems 1a - 1b:
Graph each linear function by finding
x- and y-intercepts.
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|
Practice
Problems 2a - 2b:
Graph each linear equation.
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Need Extra Help on These Topics?
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All contents copyright (C) 2001 - 2008, WTAMU and Kim Seward. All rights reserved.
Last revised on June 22, 2003 by Kim Seward. |
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