Reflecting and scaling absolute value function

- [Instructor] The graph of y

is equal to absolute value of x

is reflected across the x-axis

and then scaled vertically
by a factor of seven.

What is the equation of the new graph?

So pause the video and see
if you can figure that out.

Alright, let's work
through it together now.

Now, you might not need
to draw it visually

but I will just so that
we can all together

visualize what is going on.

So let's say that's my x-axis

and that is my y-axis.

y equals the absolute value of x.

So for non-negative values of x,

y is going to be equal to x.

Absolute value of zero is zero.

Absolute value of one is one.

Absolute value of two is two.

So it's gonna look like this.

It's gonna have a slope of one

and then for negative values,

when you take the absolute value,

you're gonna take the opposite.

You're gonna get the positive.

So it's gonna look like this.

Let me see if I can draw
that a little bit cleaner.

This is a hand drawn
sketch so bear with me

but hopefully this is familiar.

You've seen the graph

of y is equal to absolute
value of x before.

Now, let's think about the
different transformations.

So first, they say is
reflected across the x-axis.

So for example,

if I have some x value right over here,

before, I would take
the absolute value of x

and I would end up there

but now we wanna reflect across the x-axis

so we wanna essentially get
the negative of that value

associated with that corresponding x

and so for example, this x,

before, we would get
the absolute value of x

but now we wanna flip across the x-axis

and we wanna get the negative of it.

So in general, what we are doing

is we are getting the negative
of the absolute value of x.

In general, if you're
flipping over the x-axis,

you're getting the negative.

You're scaling the expression
or the function by a negative.

So this is going to be y
is equal to the negative

of the absolute value of x.

Once again, whatever absolute value of x

was giving you before for given x,

we now wanna get the negative of it.

We now wanna get the negative of it.

So that's what reflecting
across the x-axis does for us

but then they say scaled
vertically by a factor of seven

and the way I view that is if
you're scaling it vertically

by a factor of seven,

whatever y value you got for given x,

you now wanna get seven times the y value,

seven times the y value

for a given x.

And so if you think
about that algebraically,

well, if I want seven times the y value,

I'd have to multiply this thing by seven.

So I would get y is
equal to negative seven

times the absolute value of x

and that's essentially
what they're asking,

what is the equation of the new graph,

and so that's what it would be.

The negative flips us over the x-axis

and then the seven scales
vertically by a factor of seven

but just to understand
what this would look like,

well, you multiply zero times seven,

it doesn't change anything

but whatever x this is,

this was equal to negative x

but now we're gonna get
to negative seven x.

So let's see, two, three,
four, five, six, seven

so it'd put it something around that.

So our graph is now going to look,

is now going to look like this.

It's going to be stretched
along the vertical axis.

If we were scaling vertically

by something that had an
absolute value less than one

then it would make the graph less tall.

It would make it look,

it would make it look wider.

Let me make it at least look
a little bit more symmetric.

So it's gonna look something,

something like that but the key issue

and the reason why I'm
drawing is so you can see

that it looks like it's
being scaled vertically.

It's being stretched in
the vertical direction

by a factor of seven and the
way we do that algebraically

is we multiply by seven

and the negative here is what
flipped us over the x-axis.