Shifting and scaling parabolas

Here I've drawn the
most classic parabola,

y is equal to x squared.

And what I want to do is think
about what happens-- or how can

I go about shifting
this parabola.

And so let's think about
a couple of examples.

So let's think about
the graph of the curve.

This is y is equal to x squared.

Let's think about what
the curve of y minus k

is equal to x squared.

What would this look like?

Well, right over here, we
see when x is equal to 0,

x squared is equal to 0.

That's this yellow curve.

So x squared is equal to y,
or y is equal to x squared.

But for this one, x
squared isn't equal to y.

It's equal to y minus k.

So when x equals a
0, and we square it,

0 squared doesn't get us to y.

It gets us to y minus k.

So this is going to
be k less than y.

Or another way of thinking
about it, this is 0.

If it's k less than y, y must
be at k, wherever k might be.

So y must be at k,
right over there.

So at least for this
point, it had the effect

of shifting up the y value by k.

And that's actually true
for any of these values.

So let's think about x
being right over here.

For this yellow curve,
you square this x value,

and you get it there.

And it's clearly not
drawn to scale the way

that I've done it
right over here.

But now for this
curve right over here,

x squared doesn't cut it.

It only gets you to y minus k.

So y must be k higher than this.

So this is y minus k. y
must be k higher than this.

So y must be right over here.

So this curve is essentially
this blue curve shifted up

by k.

So making it y minus k is equal
to x squared shifted it up

by k.

Whatever value this
is, shift it up by k.

This distance is a constant
k, the vertical distance

between these two parabolas.

And I'll try to draw
it as cleanly as I can.

This vertical distance
is a constant k.

Now let's think about shifting
in the horizontal direction.

Let's think about what happens
if I were to say y is equal to,

not x squared, but
x minus h squared.

So let's think about it.

This is the value you would get
for y when you just square 0.

You get y is equal to 0.

How do we get y
equals 0 over here?

Well, this quantity right
over here has to be 0.

So x minus h has to be 0,
or x has to be equal to h.

So let's say that h
is right over here.

So x has to be equal to h.

So one way to think about
it is, whatever value you

were squaring here
to get your y,

you now have to have
an h higher value

to square that same thing.

Because you're going
to subtract h from it.

Just to get to 0,
x has to equal h.

Here, if you wanted to square
1, x just had to be equal to 1.

So here, let's just say,
for the sake of argument,

that this is x is equal to 1.

And this is 1 squared,
clearly not drawn to scale.

So that would be 1, as well.

But now to square 1, we don't
have to just get x equals 1.

x has to be h plus 1.

It has to be 1 higher than h.

It has to be h plus 1 to
get to that same point.

So you see the net
effect is that instead

of squaring just x,
but squaring x minus h,

we shifted the
curve to the right.

So the curve-- let me do this in
this purple color, this magenta

color-- will look like this.

We shifted it to the right.

And we shifted it
to the right by h.

Now let's think of another
thought experiment.

Let's imagine that-- let's
think about the curve

y is equal to
negative x squared.

Well, now whatever the
value of x squared is,

we're going to take
the negative of it.

So here, no matter what
x we took, we squared it.

We get a positive value.

Now we're always going
to get a negative value

once we multiply it
times a negative 1.

So it's going to look like this.

It's going to be a
mirror image of y

equals x squared reflected
over the horizontal axis.

So it's going to look
something like that.

So that's y is equal
to negative x squared.

And now let's just imagine
scaling it even more.

What would y equal
negative 2x squared?

Well, actually, let
me do two things.

So what would y equals
2x squared look like?

So let's just take
the positive version,

so y equals 2x squared.

Well, now as we
square things, we're

going to multiply them by 2.

So it's going to
increase faster.

So it's going to look
something like this.

It's going to be
narrower and steeper.

So it might look
something like this.

And once again, I'm just
giving you the idea.

I haven't really
drawn this to scale.

So increasing it by a factor
will make it increase faster.

If we did y equals
negative 2x squared,

well, then it's going to get
negative faster on either side.

So it's going to look
something like this.

It's going to be the mirror
image of what I just drew.

So it's going to be a narrower
parabola just like that.

And similarly-- and I know that
my diagram is getting really

messy right now--
but just remember

we started with y
equals x squared,

which is this curve
right over here.

What happens if we did
y equals 1/2 x squared?

I'm running out of
colors, as well.

If we did y equals
1/2 x squared,

well, then the thing's
going to increase slower.

It's going to look the same,
but it's going to open up wider.

It's going to increase slower.

It's going to look
something like this.

So this hopefully
gives you a sense

of how we can shift
parabolas around.

So for example, if I have-- and
I'm doing a very rough drawing

here to give you the
general idea of what

we're talking about.

So if this is y
equals x squared,

so that's the graph
of y equals x squared.

Let me do this in a color
that I haven't used yet--

the graph of y minus k is equal
to A times x minus h squared

will look something like this.

Instead of the vertex
being at 0, 0, the vertex--

or the lowest, or
I guess you could

say the minimum or
the maximum point,

the extreme point in the
parabola, this point right

over here, would be the maximum
point for a downward opening

parabola, a minimum point for
an upward opening parabola--

that's going to be shifted.

It's going to be shifted
by h to the right and k up.

So its vertex is going
to be right over here.

And it's going to be scaled
by A. So if A is equal to 1,

it's going to look the same.

It's going to have
the same opening.

So that's A equals 1.

If A is greater than 1, it's
going to be steeper, like this.

If A is less than 1
but greater than 0,

it's just going to be
wider opening, like that.

Actually, if A is 0, then it
just turns into a flat line.

And then if A is negative
but less than negative 1,

it's kind of a broad-opening
thing like that.

Or I should say greater
than negative 1.

If it's between
0 and negative 1,

it will be a broad-opening
thing like that.

At negative 1, it'll
look like a reflection

of our original curve.

And then if A is less
than negative 1--

so it's even more
negative-- then it's

going to be even a
steeper parabola that

might look like that.

So hopefully that
gives you a good way

of how to shift and
scale parabolas.