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Current time:0:00Total duration:11:31

CCSS.Math:

Let's graph another rational
function, because you really can't get enough
practice here. So let's say we have y is equal
to x over x squared minus x minus 6. So the first thing we might want
to do is just factor this denominator so we can identify
our vertical asymptotes, if there are any. So what two numbers when I
take their product I get negative 6 and if I add them
up I get negative 1? So they have to be of
different signs. So one's going to be a plus and
one-- let me write my x's a little bit neater than that--
so one is going to be a positive and one is going
to be a negative. A 2 and a 3 seem to be pretty
close, because they're one apart, and I'm going to subtract
the larger number because when I add them,
I get a negative. So x minus 3 times x plus
2 seems to work. That gets negative 6. Negative 3x plus 2x. negative 3 times x
plus 2 times x is negative x, so that works. So this is equal to x over
x plus 2 times x minus 3. And like we saw in the last
video, since these expressions, since the x plus
2 doesn't cancel out with anything in the numerator and
the x minus 3 doesn't cancel out with anything in the
numerator, we know that these can be used to find our
vertical asymptotes. The vertical asymptotes are when
either that term is equal to zero or when that term is
equal to zero, because at those points, our equation
is undefined. So this is equal to zero when x
is equal to negative 2, and this is equal to zero when
x is equal to positive 3. And you could try it out here. If x is equal to negative 2 or
positive 3, you're going to get a zero in the denonminator, y will be undefined. So vertical asymptotes at x
is equal to negative 2. So there's a vertical asymptote,
a vertical asymptote right there. Another vertical asymptote
is x is equal to 3. One, two, three. There is our other vertical
asymptote. Now let's think about horizontal
asymptotes, or if there are any. So what happens as x gets
super-positive or super-negative? And as we said before, you
just have to look at the highest degree term on the
numerator and the highest degree term on the
denominator. Now, notice the highest degree
term on the denominator is x squared, while the highest
degree term on the numerator is only an x. So when x gets really large,
what's going to happen? You could imagine, this is going
to be like a million over a million squared, which
is still one over a million. These terms over here
don't matter much. But this term right here is
going to grow faster than everything. This is an x squared term. As x gets large, it's going
to get way larger than everything, including this
term on the top, so it's essentially going
to go to zero. When the denominator just gets
bigger, faster than the numerator as you're going
to approach zero. So we have a horizontal
asymptote at y is equal to 0. I could draw it as a dotted
line over our x-axis. So that right there is the
line, y is equal to 0. Once again, we identify that
looking at the highest degree term there. The denominator has a
higher-degree term, so it's going to grow faster
than the numerator. You could try it out
on your calculator. And that's true whether you
go in the super-negative direction or the super-positive
direction. This thing is going to overwhelm
this thing up here, the denominator grows faster
than the numerator, which essentially we're going
to approach zero. You're going to get smaller
and smaller fractions. Just remember, 1/10 and then--
let me actually just-- as x gets larger and larger
and larger, what's going to happen? Let me just show you
on my calculator. Let's say x is equal to 10. 10 divided by 10 squared minus
10, and normally you wouldn't have to do this. I just really want to show
you the intuition. Whoops! I'm not trying to graph. Let me exit from here. So if we have 10 over 10 squared
minus 10, once again, you normally wouldn't
have to do this. I just want to show you,
give you the intuition. Let me put some parentheses
there. Let me put some parentheses
over here. So let me insert the parentheses
there and put a parentheses over here. You get a small number. What happens if x gets
even larger? Let me make, instead of a 10,
let me make it all 100. Let me make these tens into
hundreds, into 100. Insert 100 there,
what do we get? We get even a smaller number. And if you try x is equal to
1,000, it's going to be be even smaller than that. That's because this term right
here is growing faster than every single other term. That's why our horizontal
asymptote is y is equal to 0. Now, the last thing we want to
do, we've drawn all of our asymptotes, is just try
out some points. So let's draw like a
little table here. There's our table. When x is equal to
0, what is y? x is 0, we have 0 over
all of this. 0 minus 6, 0 over negative
6 is just 0. When x is equal to-- I don't
know, let's just try when x is equal to 1, what do we have? We have 1 over-- I'll
write it here. 1 over 1 squared minus 1. Now that's just 0, so
we have negative 6. When x is equal to negative
1, what do we have? When x is equal to negative
1, we have negative 1 over negative 1 squared, which
is 1 minus negative 1. So that's plus 1-- right, minus
negative 1-- minus 6. So what is this right here? This is negative 1, so this is
going to be negative 1 over 2 minus 6 over negative 4. This is going to be equal to
1/4, so we're going to get a positive value. So we have-- let me draw this--
negative 1, we're at 1/4 right here. That's about right there. I'll do it in a darker color. We had the point 0, 0, and then
at x is equal to 1, we had negative 1/6. So you could keep graphing more
and more points, but it looks like as we approach this
vertical asymptote from the right, we go to positive
infinity. And that should make sense. Let's see, if we were to put
in-- we're approaching negative 2 from the right. So if you were to put in
negative 1.9999999, this term is going to be a very small
positive number. This term's going to be
a negative number. This term's going to
be negative number. The negatives cancel out. You have a very small positive
number in the denominator. 1 over that gives you a
very positive number. Now, as we approach the other
vertical asymptote from the left, we're going to
go super-negative. My gut tells me that because
when I tried x is equal to 1, I already went to a
negative value. But you could imagine if
you did 2.99999, right? Let me draw that a little
bit better. You get the idea. If x is equal to 2.999, so we
get really close to the asymptote, this is going to be
positive, this is going to be negative, that's going to be
positive, and this is going to be a small number. So you're going to have 1 over
a very small negative number, which is a very, I guess,
negative number. It's a negative of 1 over a very
small number, so you're going to approach negative
infinity. Now, let's try some point out
here to see what happens. So what happens when
x is equal to 4? When x is equal to 4, you have
4 over 16 minus 4 minus 6. What is that? That's 16 minus 10. That is 6. So this is equal to 4/6,
which is equal to 2/3. So the point 4, 2/3 is here,
so one, two, three, four. 2/3, just like that. So that gives me the sense,
look, I have to approach this horizontal asymptote as we go
further and further out. We're going to probably
approach positive infinity like this. Let me draw it a little
neater than that. Like that. You get the idea. Then over here, we're going to
get closer and closer to our horizontal asymptote as
we approach infinity. This should be a
smoother-looking curve right around there. I'm making a mess here. This should be a
smoother-looking curve. You get the idea, I think. Now, let's see what happens when
x is equal to negative 3. So when x is equal to negative
3, we have negative 3 over negative 3 squared, which is 9
minus negative 3, so that's plus 3 minus 6. So what is this equal to? This is equal to negative 3
over-- this is 12 minus 6 over 6, right, which is equal to
negative 1/2, So negative 3, negative 1/2. Negative 1/2 is right there. So we're going to approach
this asymptote as we get really negative. And we're probably going to go
straight down like that as we approach this vertical asymptote
right there. And you could try more points
if you don't believe me. But let us graph it just to
verify it for ourselves. So our equation is x divided by
x squared minus x minus 6. And let's graph it. And there you go. There you go. All right, looks pretty good. Our asymptote is zero,
we go down. Vertical asymptote, bam! Go up there, then we go back
down here, then we go just like that again. So once again, this looks just
about exactly what we got. Obviously, the graphing
calculator, it kind of a pitters out as you get close
to these values and it does weird things, but it has
the same general shape. We could actually close the
range a little bit if we want, if we want to graph it. Let's make our x minimum
value, let's make it 5. And let's take our x maximum
value, let's make that 5 as well. We're kind of zooming
in a little bit. So let's graph it now. Bam! bam! There you go! It's the same shape as we
graphed right here. Hopefully, you found
that satisfying.