If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:21:06

CCSS.Math:

In this video, we're going
to see if we can graph a rational function. A rational function is just
a function that has an expression on the numerator
and the denominator. It has a polynomial in the
numerator-- Let's see, we have x squared over-- and another
polynomial in the denominator --x squared minus 16. We could obviously graph this by
just trying out a bunch of points and then connecting
the dots. That's what a calculator would
do for us, a graphing calculator. But what we want to do is,
before we try out some points to kind of fill in the gaps, I
want to understand the basic structure of this graph first. And to understand that, I want
to see what happens as x gets really big, so x gets really
big, or x gets really, really small, as x goes in the
negative direction. Or another way we could think
about it, I want to understand what happens as the magnitude of
x, or the absolute value of x, becomes really big as it
approaches really, really bigness, or as it approaches
infinity. So when the size of x
approaches infinity. Which essentially is saying,
as x goes really far in the positive direction, or x goes
really far in the negative direction, what is going
to happen to the value of this function? So let's get out a calculator. Won't use the graphing part of
it just yet, but let's just try out some values. What happens when x
is equal to 10? It's going to be the same thing
as when x is equal to negative 10, because when you
put a negative 10 here, you square it, you get 100,
just like 10. Same here, negative 10, you
square it, you get the same thing as a positive 10. So whether you go in the super
high positive direction or the super low negative direction,
as you approach positive or negative infinity, you're going
to approach the same thing because you're squaring
the values. But let's try out some values. If I get 10 squared divided
by 10 squared minus 16, I get 1.19. Now what happens if x gets
a little bit bigger? This is x is equal to 10. What happens when x
is equal to 100? We have 100 squared divided
by 100 squared minus 16. I'm getting even closer to 1. When x was 10, around here,
we're getting y is 1.19. When x is 100, 100 squared
over 100 squared minus 16, y is 1.0016. Just for fun, let's try 1,000. So it's 1,000 squared divided
by 1,000 squared minus 16. And we're even closer to 1. So as the size of x gets larger
and larger and larger, our y gets closer
and closer to 1. And that would also be true
if this was a negative 10, because negative 10 squared over
negative 10 squared minus 16 is going to be the
exact same thing. Because the negative, when
you square it, is going to be a positive. It's going to be the same thing
as 10 squared, same thing over here. So whether x gets really big or
x gets really small, we're going to be approaching
y is equal to 1. You could try it with a million
if you want, and you're going to get a number
even closer to 1. So as the size of x approaches
infinity, the absolute value of x, or the distance from the
origin, approaches infinity, y is approaching 1. Or another way to think about
it is, the graph of this function is going to approach
the line y is equal to 1. So let me graph the line
y is equal to 1. So I'll do it in a dotted line
because this isn't the graph of our function, but this is
a line that our function is approaching. So that is the graph
of y is equal to 1. Now, this idea of a function,
or the graph of a function, approaching a line but never
quite touching it. So this is going to get closer
and closer and closer to this line, y equal to one, in that
direction, but never quite getting close enough to it. It'll approach 0, its distance
from this y equals 1, but it'll never quite get there. This line that the graph
is approaching is called an asymptote. And it'll be even more
clear once I actually graph the function. We're going to work up there. And since it's a horizontal
line, we call this a horizontal asymptote. This is what our graph
approaches as we go in the positive direction, or really
far in the negative direction. Let's think about some of the
other interesting things about this, about this function
right here. Well one thing that might pop
out at you is this is a difference of squares. This is x squared
minus 4 squared. So we can rewrite this as
x squared over x plus 4 times x minus 4. So what's going to happen here,
as x approaches either positive 4, or x approaches
negative 4? Well, first of all, try
those values out. If x is equal to 4, what
is going to happen? This expression right here, this
term right here, is going to be equal to 0. And we're going to
be dividing by 0. We cannot do that. Similarly, if x is equal
to negative 4, we'd be dividing by 0. This expression, right here,
is going to be equal to 0. We can't do that. We could say that this function
is undefined at x is equal to plus or minus 4. It can't equal those values
because we'd be dividing by 0 in either one of those
circumstances. Now, what happens as we
approach those values? What happens as x approaches
negative 4? Let's just do that
one for fun. What happens as x approaches
negative 4? Let's say we're approaching it
from the negative direction. So let's try it out
in our calculator. So let's say we want to go from
the negative direction. So let's start with
negative 4.1. So if we have negative 4.1
squared divided by negative 4.1 squared minus 16,
what do we get? We get 20.75. So we get this number,
whatever. Let's see if we get even
closer to negative 4. So let me just get
that entry there. So let's get a little bit
closer to negative 4. So instead of negative 4.1,
let's do negative 4.01. So let me insert a
negative 4.01. And then over here, this
is negative 4.01, and see what it is. Now we went to 200, so
we're getting to larger and larger values. Let's try negative 4.001. Let's try that out. Whoops, that's not what
I wanted to do. I wanted to do that. So let's try. No that's not what
I want to do. Let's see. So we want to go to, instead of
4.01, I want to do 4.001, and over there, negative
4.001. And what do we get? We get 2,000. So as we get closer and closer
to negative 4 from the negative direction, we're
approaching larger and larger, super larger numbers. And you can try it, if it's
4.0000001, it's going to get to smaller and smaller numbers--
or sorry, larger and larger numbers here. If you do 4.001, it's probably
going to be 20,000. And then if you add
another 0 here. So as we get closer and closer,
it's getting to larger and larger numbers. So we could say, as x approaches
negative 4, we could say y is approaching
infinity. It's getting to a larger and
larger and larger value. But we can't ever quite get
to x is equal to 4. It's undefined there. That will make the denominator
here equal to 0. So what we want to do here is,
we can never quite equal x equal negative 4. So let me see, x is equal
to 1, 2, 3, 4. We can never quite get to x
is equal to negative 4. Let me draw x is equal to
negative 4 as a dotted line, right there. That is x is equal
to negative 4. We can never quite get there,
but as we approached it from the negative side, as we had
4.1, then 4.01, we went to larger and larger values. And we also know that as we went
on the left-hand side, as we go to larger and larger x
values, that y will get closer and closer to 1. So you have a general sense of
what this part of the graph will look like. This part of the graph
is going to look something like that. As x gets to super negative
numbers, it gets closer and closer to 1, as x gets closer
and closer to negative 4 from the negative direction, it's
going to go closer and closer to infinity. You're going to get closer and
closer to a very-- It's going to get larger and larger,
I guess, is an easy way to say it. Now, just like x equal negative
4, x equals 4 will also be a point where the
graph is undefined. So let me graph that here. 1, 2, 3, 4. Right here. Right over here. x is equal to 4. And, once again, what happens
as we approach x equals 4, let's say from the positive
direction? So as x approaches 4 from the
positive direction, what's going to happen? So this is like trying out x
is equal to 4.01, or x is equal to 4.001, or x
is equal to 4.0001. So we're just getting closer and
closer and closer to x is equal to 4. Well, these values are the exact
same values that we just tried on our calculator, except
they are the negative version of them, right? And we already saw that, just
the way that this function is set up, the negative numbers,
they get squared, so whether you take the negative or the
positive x values, it's going to be the same thing. This graph is symmetric. When x is equal to negative
5, it's the same thing as x is equal 5. When x is equal to negative
10, it's the same thing as x equals 10. So the same thing is
going to happen. You could try it out with your
calculator, if you like. If you try out these values,
you're going to see, as we get closer and closer to 4, we're
going to approach larger and larger numbers. These same numbers over here. So the graph over here, as we
get closer and closer to 4, we're going to approach larger
and larger numbers. And then here, as x gets larger
and larger and larger, we saw over here, we had these
horizontal asymptotes, y gets closer and closer to 1. So just like we called this a
horizontal asymptote, these values-- or you can even view
these vertical lines: x is equal to negative 4 and x is
equal to 4 --we call these vertical asymptotes. These are lines, asymptotes,
once again, they are lines that the graph approaches,
but never quite touches. So that's what's
going on here. And then we can think about
what's happening to the graph inside of here. So you could think of it
in a couple of ways. You could say, well, what
happens as x approaches 4 from the negative direction? So let's try that out, from
the negative direction. So what happens if you do 3.9
squared divided by 3.9 squared minus 16? You get negative 19.25. Now what happens
if we do 3.99? So let me put another 9 here. So we're going to get closer
and closer to 4, and we're going to do it from
the left-hand side as we approach 4. So insert another 9 here. So even more negative. So let's just do one more. So we're going to be
even more negative. So let me make it 3.999. Get even closer to 4. You're getting even
more negative. And this is also going to be
true if we did negative 3.9, or negative 3.99, or negative
3.999, because when we square it, the negatives and the
positives are the same thing. You square negative 1,
you get a positive 1. So as we approach 4 from-- you
go 3.9, 3.99, we get closer and closer to 4 --we're
getting more and more negative numbers. We approach negative infinity. So as we approach-- let
me just graph it here. As we approach from this
direction, we're going to get smaller-- want to not touch our
asymptote --it's going to look something like that. As we approach it from the
left-hand side, we're getting smaller and smaller numbers. And that's also going to be true
as we approach negative 4 from the right-hand
side, right? As we get negative 3.9,
3.99, 3.999, we're going to drop down. It's going to look something
like that. And now that we have a general
sense of what the graph is, now is a good time where
we could maybe plot a few points here. And the easiest one is, what
happens when x is equal to 0? You have 0 squared over
0 squared minus 16. So the point when x is equal
to 0, we're going to have 0 over, well, negative 16,
which is just 0. So the point 0, 0 is
on this curve. And then we could try some other
points if you like, but the general shape here
is going to look something like this. You could plot more points if
you really want to nail down exactly what the curve is doing
in between, but here is the general structure. And we tried out a lot of values
with the calculator. And I did that because I really
wanted to show you why it's dropping down like this. And if you think about it,
it makes complete sense. As you get closer and closer,
let's say you get closer and closer to 4. Either way, as you get closer
and closer to 4, this is going to become a smaller and smaller
and smaller number, because this is the difference
between x and 4. So this is becoming
a smaller and smaller and smaller number. Then, when you take 1
over that, right? You can essentially view this as
x squared over x plus 4, or times 1 over x minus 4. If this is becoming smaller and
smaller, this whole thing, 1 over a super small number,
is a super large number. So as you can imagine, you're
going to get larger and larger, and depending on whether
you are approaching from the positive or negative,
so whether this is a super small negative number or super
small positive number, that's going to flip the sign. But either way, the magnitude--
So we're getting to a very large magnitude in the
negative direction because the difference between
x and 4 on this side is negative, right? 3.9 minus 4 is 0.1. Take the inverse of
that, it's 10. So we're getting negative
numbers here. You take the inverse, you're
going to get super large negative numbers. So I really want to give
you that intuition. But the general way of being
able to graph these type of things, your first thing you
want to do is identify the horizontal asymptotes. What happens as we get very--
the magnitude of our x is very large, so super positive
values or super negative values. You could try it out on a
calculator, if you like. You literally, if you try out
the value of a million or a billion, it will kind of
give you the answer. But the way you could also think
about it is, as x gets really large, you could view
that this thing, these terms right here grow so much faster--
I mean this is just a constant term. This term doesn't
matter anymore. If this is a million
and a million, who cares about the 16? So as x gets really large,
you could say that y is approximately x squared
over x squared. These two terms dominate. You don't need to worry
about the 16 anymore. And of course, this is equal to
1, which is exactly what we got when we plugged in
really large numbers. So, in a problem like this,
where you have the same coefficient, or where you have
the same degree on the numerator and the denominator,
you look at the coefficient of those terms. So in this case,
the coefficient is 1 and 1. So our horizontal asymptote is
going to be 1 divided by 1, or y is equal to 1. If this was 2x squared over
x squared minus 16, our horizontal asymptote would
be y is equal to 2. We would approach that
line, up there. If it was a negative 2, our
horizontal asymptote would be y is equal to negative 2. So that's how you identify the
horizontal asymptotes where you have the same degree in
the numerator and the denominator. If the denominator has a
larger degree, then the denominator is going to get
larger much faster than the numerator, and your asymptote
is going to be 0. I'll show an example of
that in the future. And obviously, if your numerator
has a higher degree than your denominator, it's
going to grow way faster than your denominator, and you won't
have any asymptote. You'll just keep growing,
or keep going in the negative direction. And that's actually the case
with all of the polynomials we've seen. You can do them all
as being over 1. In which case, there was no
horizontal asymptote. Now the vertical asymptotes you
identify by essentially just factoring the denominator
and figuring out where does it equal 0. Those are the points where the
function is not defined. And I'll show you in the future,
there are some special cases where they won't be
vertical asymptotes, and I guess that special case is,
for example, if you had-- Well, I won't show you the
special case right now. I'll show you that in
a future video. But in general, if you factor
the bottom terms and they don't cancel out with anything
on the numerator, then you're going to be dealing with
a vertical asymptote. If I had another x minus 4 up
here, if my numerator was x squared times x minus 4, and
then these canceled out, and my expression simplified to
this, the equation would still be undefined at x is equal to 4,
because you would give the 0 in the denominator. But since that x minus 4 cancels
out with the x minus 4 in the numerator, it
would not have been a vertical asymptote. We'll look at that
in the future. But this equation wasn't that. So the general rule of thumb for
identifying the vertical asymptotes, factor the
denominator, figure out where the denominator equals 0, and
if those terms don't cancel out with any terms of the
numerator, then those are vertical asymptotes. And then to figure out the
behavior, I guess, within the asymptotes, you can
plot some points. You can try out some points. You can actually substitute
values for x and figure out what y is. Now just to validate that we
hopefully got the right answer, let's actually graph
our rational function. So let me turn it on. Let me graph it. And we say y is equal to
x squared divided by x squared minus 16. And let's see what we get. Nope, I just want to graph it. My range is off. Let me do my range. Let me see, x minimum value I
want for x, let's say it's negative 10. My maximum value I
want for x is 10. x scale is 1. y minimum value,
I want negative 10. y maximum value, I want 10. Then y scale, I want 1. Now let me graph it. There we go. Look at that. Just like what we drew. We have an asymptote, as x gets
really large, or x gets really, really small, that
asymptote is y is equal to 1. We have our vertical
asymptote. It graphed it because it tried
to connect the dots, but it essentially graphed our
asymptotes for us, but that wouldn't actually be
part of the graph. But as we approach 4 from
0, I guess we can say, we go super negative. As we approach negative 4 from
0, we get super negative. Because in either of those
situations, as you approach 4 from this side, this term
is going to be negative. As you approach negative 4 from
thid side, this term, right here, is going to be--
Well, this term right here is going to be positive, but then
this term right here is going to be negative. Negative times a positive,
you could play with it as you like. But we approach negative
infinity in either case. And then as x approaches
infinity, this thing asymptotes away. So hopefully you
found that fun.