Tip:
Highlight text to annotate it
X
>> In this video I show you a way to graph equations of lines.
If the equation is solved for Y. For instance, graph Y equals 2 X minus 4.
Keep in mind one reason for graphing the line is that's a visual representation
of all the ordered pairs that are solutions to that equation.
All right, so what we do is make up a little chart.
We could pick a value for X and then figure out what Y value goes along
with that X. Now there's an intermediate step which you may or may not need to do.
I'm going to put the intermediate step in.
Y is really 2 X minus 4.
So I'm going to show you how I get Y. So -- well, what can you put in for X?
Any number you want.
So I usually pick a negative number, zero, and a positive number.
But you could pick any number you want.
So let's say we're going to plug in negative 1 for X and then we're going
to plug in zero, and how about 1 and 2.
It's true that you only need two ordered pairs to draw a straight line.
But I suggest you do at least three to make sure that you haven't made a mistake.
So if you've got three points on a line that really will do it.
A fourth is just an extra check.
So what we're going to do is plug in negative 1 for X.
So I'm going have 2 times negative 1 minus 4, and I'm going to simplify that.
So negative 2 minus 4 is negative 6.
This is the ordered pair, negative 1, negative 6, for instance.
All right, let's put in zero.
So we put 2 times 0 minus 4.
You have to be careful with your arithmetic of course.
That's 0 minus 4, which is negative 4.
So we have another ordered pair, 0, negative 4.
Now if I didn't make any mistakes, then if I plot these two points and draw a line
through them that will be my graph.
Let's just do two more for practice.
This will be 2 minus 4, which is negative 2.
And plugging in 2 for X, this gives me 4 minus 4, or 0.
So I have four ordered pairs that I've shown.
So our next step would be to take these four ordered pairs
and plot them on a -- carefully -- on a graph.
And then draw the straight line through them.
Before going on I want to point out that some people don't have to do that middle step.
They simply write X, plug in negative 1, do it in their head and write what the value of Y is,
and they've got these ordered pairs in this kind of a T chart.
Sort of like a T. Or some people simply just write the ordered pairs.
Put a negative 1, put it in their head, two times negative 1 minus 4,
and write the ordered pair right next to it instead of in a T chart.
So it's really up to you how you get these ordered pairs
and how much work you need to show.
But basically we've got, whether we're looking at the T chart or we're looking
at these four ordered pairs we want to plot those on the graph.
So I want you to get out a piece of graph paper, carefully plot those points,
get something straight like a ruler, and draw the arrow -- draw the line through those points.
Here are the four points.
Next step is to draw a straight line.
After carefully drawing a straight line it doesn't hurt to pick some point
that line goes through, as long as you graph it carefully,
and make sure it also makes the equation true.
So I notice there's this other point right here which is the ordered pair 2, 3.
And convince yourself if you put 2 for X and 3 for Y that that would -- oops, I did that wrong.
That would be 3, 2.
That wouldn't have worked.
3, 2. If I put in 3 for X and 2 for Y is it true.
So my original equation here, if I put in 3 for X and I put in 2 for Y, I see that 2 is equal
to 2 times 3, 6 minus 4, yes, that's true.
So this does give me the correct graph of that line.
Let's go on to another equation, and this one has fractions.
Okay, so that scares a lot of people off.
But we can do this one just as easily.
Remember, you can pick any number you want for X.
So you could just put in 0, 1, and 2, for instance.
But if you put in, like, 1 for X, you're going to have to deal with fractions, and add,
get common denominators, et cetera.
So there's a trick to this.
And that is since you could pick any number you want
for X pick something that's easy to plug in.
Now what's going to be easy -- hmm.
Some number that 3 is going to cancel with.
You want that 3 in the denominator to cancel with something
for X. That would be any multiple of 3.
Like, 3, 6, 9, 12, et cetera.
I'm going to pick a negative number, so I'll pick negative 3.
I always plug in 0, because that's a super-easy number to put
in for X. And then how about positive 3.
Three ordered pairs is really all you need to put in.
So let's do this.
We're going to plug in negative 3 for X. Now, be very careful with your arithmetic here.
You have a negative times a negative as a positive,
and note because I chose a multiple of 3, the 3 cancels here.
So this is going to give me 2 plus 1.
So my Y value is going to be 3.
The next one will be easy, because any number times 0 is 0.
So this gives me 0 plus 1, or 1.
And then let's plug in positive 3.
And again, the 3s are going to cancel.
So I get negative 2 plus 1, which is negative 1.
That's really all you need.
You've got three ordered pairs, negative 3, 3; 0, 1; and 3 negative 1.
So all we need to do is plot those three points on a graph
and accurately draw a line through them.
So go ahead and try that on your own.
Put it on pause, and then check it with the graph that I'll show.
I've plotted the three points, and I noticed they're on a straight line.
If they don't lie on a straight line that means you made a mistake, and just pick another value
for X. So let's say one of those wasn't on the line.
I computed Y incorrectly.
If I would just now plug in, let's say 6, hopefully I wouldn't make the mistake twice.
I have negative 4 plus 1.
That's negative 3.
This gives you one more ordered pair, 6 negative 3.
So if one of these dots was sort of off,
hopefully you can see the three that are on the straight line.
And now the last step is to simply draw a line through it.
Remember to put the arrows on both ends of the line.
So there's our line.
Hope yours looks the same.
Be as accurate as possible.
Here's one more problem.
Y equals 1/2 X minus 3.
So remember we want to pick easy values for X because we can pick any number we want.
So since the denominator's 2, that's what you're going to look at.
You want multiples of whatever the denominator is.
2, 4, 6, et cetera.
And you're always going to plug in 0.
So I want you to plug in the values, let's say negative 2, 0, 2, and 4.
And this middle step, writing out the steps is just so you don't have
to do the arithmetic in your head.
But if you can do it in your head that's fine.
You just need to get the right Y values.
So go ahead and put it on pause and let's see what you get for your Y values.
All right, here are the steps.
We're plugging in the different values of X. Be very careful with your arithmetic.
So for instance, for this first one, you know it's going
to be a negative minus 3, and you cancel the 2.
So it's a negative 1 minus 3 is negative 4, et cetera.
So the next step would be to put these four ordered pairs on a graph
and then accurately graph the line through them.
So try that.
And this is the line going through those four ordered pairs.
You notice they're on a straight line.
So go ahead and draw that.
All right, so we've done problems where it's solve for Y. You're going to go
onto the next video to see how to deal with problems if it's not solved for Y.
And I want you to make one more observation here.
Notice in my X values I kind of up went up by 2s, from negative 2 to 0 is two spaces,
and then from 0 to 2 is two spaces, and from 2 to 4 is two spaces.
Check out what happened in the Y values when I did that.
Just went up one space, right?
Negative 4 plus 1 is negative 3.
Add one again, I get negative 2.
Add one again.
I get negative 1.
So this pattern actually continues.
If you go up the same amount, some certain amount
on the X values, like I was going up by twos.
The next one would be 6.
This one's going up by one.
What do you think the next one is going to be?
Add one to that, you get 0.
That's only if you're going up by this constant amount
on the X. All right, you're ready for lots of these?
Practice them and then go onto the next video.