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Hi, class.
Today we're going to take a look
at graphing using x and y-intercepts.
Now, an x-intercept is when you cross the x-axis.
One thing you want to notice,
is when you cross the x-axis, your y is zero.
The y-intercept is when you cross the y-axis.
And notice, when you cross the y-axis, your x is zero.
Graphing with intercepts is a very useful way to graph
when your equation is in what we call standard form.
In standard form, [Ax + By = C ]
you have some number times x plus some number times y
equals another number.
An example of an equation in standard form
would be this one where we have
2x minus 3y equals 12. [ 2x – 3y = 12 ]
When you have it like this,
what I would do is set up your table
so that you have a zero in a spot for your x
and then you're going to do another point
where the zero is in the spot for your y.
When your zero is for your x,
you're going to get the y-intercept,
because when you cross the y-axis, your x is zero.
When you have the zero for the y,
you're going to be getting your x-intercept
because when you cross the x-axis, your y is zero.
Now, we choose these values
because zeros are the easiest things to plug in,
and they make a lot of things just kind of disappear.
Your book will tell you to also find a third point,
and you can choose any x value you want for that third point.
It is fine to do the third point,
but you can really tell where the graph is going to go
just from the first two points.
But, by all means, use the third point
in order to see where your graph is going to land.
We're going to choose the x value for our third point
after we find our x and our y-intercept.
So let's find our first point
by plugging in zero for x to get the y-intercept.
When I do that, I replace this x with zero
and I get 2 times zero minus 3y equals 12. [ 2(0) – 3y = 12 ]
2 times zero is just zero [ 2 · 0 = 0 ] so that can disappear,
and we're left with an easy equation to solve, here.
Negative 3y equals 12. [ -3y = 12 ]
To solve that, I divide both sides
by negative 3. [ -3y/-3 = 12/-3 ]
And y ends up being negative 4. [ y = -4 ]
So, this ends up being negative 4, and I have the point (0,-4)
Now, let's find our next one by plugging zero in for y.
We are going to get
2x minus 3 times zero equals 12. [ 2x – 3(0) = 12 ]
3 times zero just makes that disappear. [ 3 · 0 = 0 ]
And you're left with 2x equals 12. [ 2x = 12 ]
So I'm going to divide both sides by 2. [ 2x/2 = 12/2 ]
And you end up with x equals 6. [ x = 6 ]
So, I have a point here of (6,0).
Now, we did all of the algebra down below for these,
but could have actually found these two points
just by doing the work in our head.
Because if you plug in zero for x for this point,
this term here [ 2x ] will disappear.
So you can cover it up with your finger or your hand
and then just divide the equation that's left.
Because you just have negative 3y equals 12. [ -3y = 12 ]
Divide both sides by 12, I mean, by negative 3,
and 12 divided by negative 3 is the negative 4.
That's something that we can actually just do in our head.
Just cover this up with your finger or your hand
and calculate what you need to calculate
for the equation that's left.
Similarly, for this next one, if we plug in zero for y,
this term [ -3y ] will disappear.
So just cover it up with your finger
and you have 2x equals 12 left. [ 2x = 12 ]
Divide the 12 by the 2 and you get your 6. [ 2x/2 = 12/2 ]
So, that is kind of a quick, easy way to get those points.
Now, let's determine what x value to use for our third point.
What I like to do is if I get these
and notice that these are 6 apart,
halfway between zero and 6 is 3.
If it's going to be a nice, neat number
when you take halfway between these two,
then I would use that value.
And that will typically give you
a nice, neat number for your y value.
If it's not a nice, neat number and it's a fraction,
then what I would do is take the difference here
and then maybe double it to get your next number.
We're going to use 3 because 3 is halfway between zero and 6.
However, if 3 had not been a nice, neat number,
I could have said, "Well, this is 6 apart;
let's go 6 more and plug in 12 here."
But it's a nice, neat number, so let's use 3 here.
And that's one way to make things work out nicely for you
for your third point.
So we're going to plug in 3 for x,
and then we do actually just have to calculate out
and solve the equation.
So, 2 times 3 minus 3y is going to be 12. [ 2(3) – 3y = 12 ]
And 2 times 3 is 6. [2( 3 ) = 6 ]
[ 6 - 3y = 12 ] Now we need to move the 6 over,
so we'll subtract 6 from both sides. [ 6 – 6 – 3y = 12 – 6 ]
Those cancel. [ 6 – 6 = 0 ]
We have -3y = 6. [ -3y = 6 ]
Divide both sides by negative 3. [ -3y/-3 = 6/-3 ]
And we end up with y equals negative 2. [ y = -2 ]
So we have a negative 2 here,
and the point is (3,-2) for our third point.
Let's plot those.
The first point is (0,-4).
So, (0,-4) lands right here.
The second point is (6,0), which lands over here.
And that third point we found was (3,-2)
and notice that they line up really nice.
And so we just connect them, and there's our line.
So that's how we go ahead and graph using intercepts.
When your equation is in the standard form [ Ax + By = C ] here,
making a table using your zero for your x
and then zero for your y to find your y-intercept
and your x-intercept will be very, very helpful for you
And make it much, much easier to graph equations in this form.