Graphs of rational functions: horizontal asymptote | Algebra II | High School Math | Khan Academy

- [Voiceover] Let f of x
equal negative x squared

plus a x plus b

over x squared plus c x plus d

where a, b, c, and d
are unknown constants.

Which of the following is a possible graph

of y is equal to f of x?

And they told us dashed
lines indicate asymptotes.

So this is really interesting here.

And they gave us four choices.

We see four of them,
three of them right now.

Then if I scroll over bit over,

you can see choice D.

And so I encourage you to pause the video

and think about how we can figure it out

because it is interesting

because they haven't
given us a lot of details.

They haven't given us
what these coefficients

or these constants are going to be.

All right, now let's think about it.

So one thing we could think about is

horizontal asymptotes.

So let's think about what happens

as x approaches positive
or negative infinity.

Well, as x, so as x approaches

infinity or x approaches

negative infinity, f of x.

F of x is going to be
approximately equal to.

Well, we're going to look
at the highest degree terms

because these are going to dominate

as the magnitude of x,

the absolute value of
x becomes very large.

So f of x is going to be
approximately negative x squared

over x squared which is equal to negative

or we could another way to think about it.

This is the same thing as negative one.

So f of x is going to approach,

f of x is going to approach negative one.

In either direction, as
x approaches infinity

or x approaches negative infinity.

So we have a horizontal asymptote

at y equals negative one.

Let's see, choice A here,

it does look like they
have a horizontal asymptote

at y is equal to negative one

right over there.

And we can verify that

because each hash mark is two.

We go from two to zero

to negative two to negative four.

So this does look like
it's a negative one.

So just based only on
the horizontal asymptote,

choice A looks good.

Choice B, we have a horizontal asymptote

at y is equal to positive two.

So we can rule that out.

We know that a horizontal asymptote

as x approaches positive
or negative infinity

is at negative one,

y equals negative one.

Here, our horizontal asymptote is

at y is equal to zero.

The graph approaches,

it approaches the x axis

from either above or below.

So it's not the horizontal asymptote.

It's not y equals negative one.

So we can rule that one out.

And then similarly, over here,

our horizontal asymptote is
not y equals negative one.

Our horizontal asymptote
is y is equal to zero

so we can rule that one out.

And that makes sense

because really they only
gave us enough information

to figure out the horizontal asymptote.

They didn't give us enough information

to figure out how many roots

or what happens in the interval

and all of those types of things.

How many zeros and all that

because we don't know what the actual

coefficients or constants
of the quadratic are.

All we know is what happens as
the x squared terms dominate.

This thing is going to
approach negative one.

And so we picked choice A.