- [Instructor] The graph of y is equal to
the absolute value of x is reflected across the x-axis
and then compressed vertically by a factor of 8/3.
What is the equation of the new graph?
All right, so let's think about this step-by-step.
If I start, and I'm just going to draw
some quick hand-drawn sketches here.
So that's my x-axis.
That is my y-axis.
If we're talking about the graph of
y is equal to the absolute value of x,
that looks like this.
In the first quadrant, it looks like y equals x,
and in the second quadrant,
it looks like y equals negative x,
'cause the absolute value of a negative number
is its opposite.
So let me make it look a little bit more
symmetric than that.
So it looks something,
something like that.
That's what the graph of y is equal to the
absolute value of x looks like.
Now, they're asking us to do what amounts to
two different transformations.
The first one is they want to reflect it.
They want to reflect across the x-axis,
so they want us to flip it across the x-axis like this.
So instead, it looks like this.
So that graph, that equation that describes this graph,
well, this is going to be the opposite.
Whatever y you were getting on this orange graph,
you're going to get the negative of that.
You're going to get y is equal to the negative
of the absolute value of x.
If this doesn't make intuitive sense to you, try it out.
In the orange graph, when x is equal to, let's say, two,
well the absolute value of two is two.
The absolute value of negative two is two.
Now, we want to take the absolute value
but then take the negative of it.
This thing stays non-positive the entire time,
so the absolute value of two is two,
but we want the opposite of that.
We want negative two.
Absolute value of negative two is two,
but we want the opposite of that.
They didn't ask us to just reflect across the x-axis.
They then want us to compress vertically
by a factor of 8/3.
So let's think about this a little bit.
Compress vertically by a factor of 8/3.
If they said stretch vertically by a factor of 8/3,
then I would just multiply this by 8/3.
Sometimes it's helpful to think of this
in terms of a mixed number.
8/3 is the same thing as 2 2/3.
If you were to stretch by 2 2/3, you would get taller.
It would get something like this.
If you are compressing,
then it's going to look something like this.
If you're compressing vertically,
you can think about it as being stretched horizontally.
If you're compressing by a factor,
you should multiply by the reciprocal of that factor.
Think about it, if you were compressing by three,
you would multiply by one-third.
If you were compressing vertically by 8/3,
that means whatever y you would have gotten,
you multiply that times a reciprocal of 8/3.
Another way you could think about it,
you could divide that by 8/3.
If we want to get this thing right over here,
this is going to be y is equal to
the negative absolute value of x.
Since we're compressing by 8/3,
we would divide that by 8/3.
Another way to think about it,
dividing by 8/34 is the same thing
as multiplying by the reciprocal.
So negative 3/8 times the absolute value of x.
When we look at our choices, we see that it is
that choice right over there.
I really want to stress this point,
because I think it can get a little bit confusing.
The reason why I multiplied that by the reciprocal,
they were saying compressing vertically.
If we said stretching vertically,
we would just multiply by 8/3,
but since we're compressing,
we would divide by 8/3 or multiply by its reciprocal.