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>> This is part four of parabolas.
And we go over graphing parabolas in this form, f of x equals plus
or minus x minus h squared plus k
and we graph the following three parabolas.
In parts one to three of parabolas we discussed how
to graph functions of the following five forms.
We talked about when a parable went, opened upward or downward,
that depended on the coefficient of x squared.
When it was positive it went up.
When it was negative it went down.
And we talked about the vertex and the axis
of symmetry of the parable.
All right.
So the last type we did was f of x equals x minus h squared,
and we're just going to add on and say what would it be
if it was of the form a minus sign in front
of it, x minus h squared.
Well the basic difference here is
that it will be opening downward instead of upward.
Again, because if you square this out you'd have a minus sign
in front of the x squared.
And the other thing we're going to look at is what
about if we had something
like x minus h squared plus k. This is a shift
of both the horizontal and the vertical axis.
Remember for this one here everything was shifted left
or right, depending on h. And when we add a k,
everything was shifted up or down.
So this is going to be both the horizontal and vertical shift.
So let's first do the one in red here.
Let's do a problem like that.
All right.
So let's say you were going to graph f
of x equals -x plus 1 squared.
Well the axis of symmetry, remember what we do
if we have the x plus 1 in parentheses squared?
We take that and set it equal to 0
and that is our axis of symmetry.
So this is our axis of symmetry at x equals -1.
Now if that's out axis of symmetry,
then our vertex is going to have that value
of x for the x coordinate.
So if we put in -1 for x, what would y be?
Well you'd have -0 squared, well the opposite of 0 is 0.
So -1, 0 is our vertex point.
Now what about this minus sign in front?
What does that tell us?
That tells us it opens downward.
[ Silence ]
Opens, opens down.
So what you could do is turn your graph paper
over if you want, if you don't like going down.
But then I just remember the usual ordered pairs
for y equals x squared.
And the basic ones are 0, 0.
1, 1. 2, 4.
And 3, 9. That's when y equals x squared.
So if I go over, so 0, 0 would be right here, right?
So then I go 1, 1.
But remember we're going to go down one.
Now if I go over two I'm going to go down four,
over three I'm going to go down nine.
Now because of this axis of symmetry,
I have a point on the opposite side.
And then we can connect.
And try to make a smooth a graph as possible,
and this goes forever.
Now that's one way of doing it.
I find that the easiest way to do it.
Another way to do it is you could certainly plug
in several values for x
until you get enough ordered pairs to get this picture.
So another way is to simply make a chart.
You could pick, you know, -3, -2, -1, 0, 1, 2, 3, 4,
whatever you want to put in.
But you're going to have to have enough points
to make sure you get the basic idea.
Often what I do is when I first look at this,
I make a little picture.
I know that the vertex is going to be -1, 0.
So I just say okay, I know this is going to be a -1, 0.
And I know it's going to go down.
So I get the basic picture before I even do the real thing.
And that might be helpful to you.
All right, let's look at the next type.
Okay. So now we're going to graph f
of x equals x minus 2 squared, minus 4.
So if you have it in this form, the x minus 2 squared,
if you want to get the axis of symmetry, you simply set
that equal to 0, x minus 2 equals 2 means x equals 2.
So the vertex point is going to have 2 for the x coordinate.
Now what happens when you put 2 in here?
Can you do that in your head?
So you're going to have, putting in 2 for x, 0 squared.
That's what it should always be,
0 squared because you set it equal to 0, right.
So it's going to be 0 minus 4 which is -4.
So there's our vertex point.
So here's a rough idea of what this is going to look like.
You're going to have this vertex point 2, -4.
And now is it going to go up or down?
Well there's a positive sign in front of here, right?
Not a negative.
So this is going to go up.
So you just know it's going to go up.
Okay. So now we're going to get out graph paper and do it.
And try this on your own first.
You can do it by plotting a bunch of ordered pairs,
or the method I just showed you in the previous problem.
Okay. So here's how I do it.
I find my vertex point, or you could do the axis
of symmetry first, it doesn't matter.
Here's the axis of symmetry.
Always dot that line, because that's where the symmetry is.
And where on that is the vertex?
It's at 2, -4.
Now you could put the 2, -4 first
and then just do the dotted line vertically.
It's up to you.
Now I simply do the same sort of order of pairs
for the y equals x squared, but it's sort
of like I'm pretending that's my vertex.
If you want you can dot this line too to say oh I'm going
to pretend that's really where the axis, the x in my axis is.
Although it's not really the real x in my axis.
I'm going to go over one and up one.
Over two and up four.
Over three and up nine.
And then there's going
to be corresponding points on the other side.
And there we are.
This should be our graph.
Now keep in mind, this is the axis symmetry, the x equals 2.
This is not anything except for you to sort of think
about as a pretend x axis, and sometimes we call
that the x prime axis, so we say,
realize that's all we're saying.
That does not mean x to the first power.
Okay. So how about trying one more.
This is a little bit trickier.
Because notice I've got this thing
in parentheses the x minus 3 squared.
I have this plus x, and I also have a minus sign out in front.
That minus sign out in front is going to tell you
if it opens upward or downward, that's my hint.
So go ahead and try this on your own, either by putting
in a bunch of ordered pairs, or sort of doing
that little fake wherever you started
from your vertex in figuring it out.
Okay so here's how I do it.
To axis of symmetry, I'm going to put
that as AOS, axis of symmetry.
Well you said x minus 3 equaled a 0, that will be x equals 3.
And that means the vertex is going to be 3.
Well what happens when you put in 3?
This always happened when you plug it in.
That will give you 0, so it really,
the ex coordinate is just going to always be that,
I mean the y coordinate
of the vertex is always going to be that number.
Hopefully you're seeing that.
And that's if it's in this form, right, with the parentheses
around the x. Or if it was just x squared.
So we've got 3, 6 for the vertex.
So I go over three and up six.
There's my vertex.
There's my axis of symmetry.
And I know it's going to go down because of this minus sign.
So in other words, I know there's a vertex
and it's going to go down.
And that's where the vertex is.
So that's the basic idea, but if you were
to get some exact points, again instead of going over one
and up one, going over one and down one.
Going over, go two and down four.
Going over three and down nine.
Corresponding points on the other side.
And this is what you should have gotten
if you did it using this method, or if you plugged
in some ordered pairs.
So here's a summarizing of our findings, graphing a parable
of this form of x equals, well it could be a plus or minus
in front of x minus h squared plus a.
If there's a positive coefficient of x squared,
in other words in front
of the parentheses is case is positive it opens upward,
negative it opens downward.
The axis of symmetry is x equals h, you could get
that by just solving, taking the part in the parentheses
and making it equal to 0 and solving for x.
And the vertex is h, k. Once you have that vertex I sort of think
of that as my new quadrant axes,
even though it's not really there.
And then I just grabbed the parable either going up or down,
remembering the easy numbers, 0, 0, 1, 1, 2, 4, 3, 9.
In other words, from y equals x squared.
We're working with this particular form of a parable
and in this form it's easy to find the vertex and axis
of symmetry, but there will be other forms we'll be working
with in later videos.