Example translating parabola

- [Instructor] Function g can
be thought of as a translated

or shifted version of f of
x is equal to x squared.

Write the equation for g of x.

Now, pause this video,

and see if you can work
this out on your own.

All right, so whenever I think
about shifting a function,

and in this case, we're
shifting a parabola,

I like to look for a distinctive point.

And on a parabola, the vertex is going

to be our most distinctive point.

And if I focus on the vertex of f,

it looks like if I shift that to the right

by three,

and then if I were to shift that down

by four,

at least our vertices would overlap.

I would be able to shift the vertex

to where the vertex of g is.

And it does look,

and we'll validate this, at
least visually, in a little bit,

so I'm gonna go minus four
in the vertical direction,

that not only would it
make the vertices overlap,

but it would make the
entire curve overlap.

So we're going to make,
we're gonna first shift to

the right

by three.

And we're gonna think about how
would we change our equation

so it shifts f to the right by three,

and then we're gonna shift down by four.

Shift down

by four.

Now, some of you might
already be familiar with this,

and I go into the intuition

in a lot more depth in other videos.

But in general,

when you shift to the right by some value,

in this case, we're shifting
to the right by three,

you would replace x with x minus three.

So one way to think about this
would be y is equal to f of

x minus three,

or y is equal to, instead
of it being x squared,

you would replace x with x minus three.

So it'd be x minus three squared.

Now, when I first learned this,
this was counterintuitive.

I'm shifting to the right by three.

The x-coordinate of my vertex
is increasing by three,

but I'm replacing x with x minus three.

Why does this make sense?

Well, let's graph the shifted version,

just to get a little
bit more intuition here.

Once again, I go into much more
depth in other videos here.

This is more of a worked example.

So this is what the shifted
curve is gonna look like.

Think about the behavior that we want,

right over here, at x equals three.

We want the same value
that we used to have

when x equals zero.

When x equals zero for the original f,

zero squared was zero.

Y equals zero.

We still want y equals zero.

Well, the way that we can do that is

if we are squaring zero,

and the way that we're gonna square zero

is if we subtract three from x.

And you can validate that at other points.

Think about what happens
now, when x equals four.

Four minus three is one squared.

It does indeed equal one.

The same behavior that you used to get

at x is equal to one.

So it does look like we have
indeed shifted to the right

by three when we replace
x with x minus three.

If you replaced x with x plus three,

it would have had the opposite effect.

You would have shifted
to the left by three,

and I encourage to think about

why that actually makes sense.

So now that we've shifted
to the right by three,

the next step is to shift down by four,

and this one is little bit more intuitive.

So let's start with our
shifted to the right.

So that's y is equal to
x minus three squared.

But now, whatever y value we were getting,

we want to get four less than that.

So when x equals three, instead
of getting y equals zero,

we want to get y equals
four less, or negative four.

When x equals four,
instead of getting one,

we want to get y is
equal to negative three.

So whatever y value we were getting,

we want to now get four less than that.

So the shifting in the vertical direction

is a little bit more intuitive.

If we shift down, we subtract that amount.

If we shift up, we add that amount.

So this, right over here,
is the equation for g of x.

G of x is going to be equal
to x minus three squared

minus four.

And once again, just to review,

replacing the x with x
minus three, on f of x,

that's what shifted,

shifted

right

by three,

by three.

And then, subtracting the four,

that shifted us down by four,

shifted

down

by four,

to give us this next graph.

And you can visualize, or
you can verify visually,

that if you shift each of these
points exactly down by four,

we are,

we are indeed

going to overlap on top of g of x.