Learning Objectives
Introduction
Tutorial
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.
y-interceptThis 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 GraphYou find the y-intercept by plugging in 0 for x and solving for y.
This is just like we showed you in Tutorial 21: Graphing Linear Equations.
If you need a review on plotting points go to Tutorial 20: The Rectangular Coordinate System.
*Inverse of mult. by -3 is div.
by -3
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.
Let’s put in an easy number x = 1:
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:
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.
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).
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:
Let’s put in another easy number x = -1:
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 = 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:
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 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:
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.
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:
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:
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 - 1b: Graph each linear function by finding x- and y-intercepts.
Practice Problems 2a - 2b: Graph each linear equation.
2a. x = 4
(answer/discussion
to 2a)
Need Extra Help on these Topics?
The following is a webpage
that can assist
you in the topics that were covered on this page:
http://www.purplemath.com/modules/intrcept.htm
This webpage goes over x- and y-intercepts.
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 July 31, 2011 by Kim Seward.
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