HELLO, Mr. Tarrou. We are going to have a series of videos and I don't think I am going
to finish them all today. We are going to be finding vertical asymptotes, horizontal
asymptotes, and slant asymptotes. Because ultimately this section in PreCalculus is
going to be coming up with, or graphing, rational equations. And rational equations have a lot
going on with their graphs. You are not going to just plot a a couple of points and see
what those look like. So the first thing you are going to do is learn as much as you can
about the graph before you start trying to consider doing a t table of points. All right,
the first thing and the easiest thing to find is if there are any vertical asymptotes. Now
that happens when you have an equation that has an x in the denominator, at least that
usually happens when you have an x in the denominator. This is because you cannot divide
by zero. So any situation where x is in the denominator and you are able to make that
denominator actually equal zero, something is going to happen there. It will usually
be a vertical asymptote. Let's take a look at a couple of examples. This is the most...
This is like a parent function. Rational functions cannot get any simpler that just simply y
equals one over x. So lets make a table of values here. We already know that we cannot
divide by zero. So the domain is all real numbers except zero. But graphically what
does this look like. Well, you could and we are going to make a table of values. We have
x values of negative two, negative one, zero, one, and two. And you can come up with others
points as well. You probably will need more than these. Let's plug those in. If you plug
in negative two you get a y value of negative one half. If you plug in negative one, one
divided by negative one is negative one. I included zero but we already know that is
undefined. So that just simply does not work. And we have the values of one, one divided
by one is one, and one divided by two is one half. Let's see if that is enough to get an
idea of what is going on here. Well I cannot have x equal to zero. I cannot divide by zero
so this graph will have a vertical asymptote at x equals zero. But what happens after that?
Well at negative two we are at negative one half. And at negative one we are at negative
one. At positive one we are at one and at positive two we are at positive one half.
So (1,1) and (2,1/2). Is that enough to let you see what is going one. Probably not. Because
you may never have seen vertical asymptotes in your graphs before. So we might need a
couple of more points, so let's add a few more. What happens as you get closer to zero
from the left and the right? Like what if happens if x is equal to negative one half?
So as we creep a little bit closer to the vertical asymptote. Well, that is going to
be y is equal to one over x is going to be one over negative one half. And as I have
explained before, like how this top divided by the bottom. You cannot really divide fractions,
that is going to flip up the bottom. One divided by negative one half is the same as one times
negative two which is negative two. Ok, so as I get closer where x is negative one half
my y is negative two. If I connect these points you can see that there is a little bit of
curvature happening. That graph is going to, if you can put in negative one fourth....negative
one tenth, you are going to see your y values become more and more negative and actually
go down to negative infinity. We will be talking about horizontal asymptotes but not in this
video. I know that this graph is going to flatten out and approach the x axis from the
bottom, but we have not talked about horizontal asymptotes yet. For now I am going to just
say take my word for it. What happens if x is positive one half? Well again. Y is equal
to one over x. That is now going to be one over one half. When you have a fraction over
another fraction you flip that bottom fraction up. You really can't divide fractions. Like
one divided by one half means that you need to change that division to multiplication
and flip that second number. Remember fractions from many many years ago. You flip the bottom
up and you get two. At one half we are at two. Again you can see this curvature happening
and the graph is going to hit.... We actually never hit the vertical asymptote, but on the
right hand side we are shooting up to positive infinity and then that graph is going down
to the right approaching zero. You can also check that with a graphing calculator if you
have one. Of course we are not doing all this work for graphing rational functions if we
can just use our calculator. The calculator will be our check. So that is y equals one
over x and a table of values trying to show you that asymptotes show up where the denominator
is equal to zero...USUALLY. What about if it were y equals one over x minus one? Where
I am taking the function and within the function I am adding the negative one. It is just a
negative one. With this one if you remember what I had up here a second ago, this is y
equals one over x. Again you cannot divide by zero so this x minus one cannot be zero,
so x cannot be one. So now instead of a vertical asymptote at zero, now my vertical asymptote
is at x equals one. The graph is going to just simply move to the right one unit. Now
if you remember your section about transformations, you will know that if you have a plus or minus
inside your math function that is a translation left or right. So when you see a negative
one inside this basic function, it was one over x, and now it is one over x minus one
with just that subtraction inside the function, you should expect that horizontal shift of
one to the right. So again where the denominator equals zero, I did not do a t table because
it was just a shift, and our vertical asymptote now is at x equals one instead of x equals
zero. Ok, so one more example like that. This time I am going to cheat and use my graphing
calculator to give me a table. But we are going to have y equals x plus seven over x
plus two. Again I told you you cannot divide by zero. Now if I can make this denominator
equal zero that is probably be a vertical asymptote. So x plus two cannot equal zero,
so x cannot equal negative two. I contend that at negative two, there is going to be
a vertical asymptote because I cannot divide by negative two. If I use my handy dandy calculator
here and graph go to my current diagrams, why yes this graph is going to approach and
go down against the left side of the asymptote. And it will come up and look something like
this approaching and shooting up towards infinity on the right side of the asymptote. You might
be going hey you are not supposed to be using a graphing calculator t tables. No, but so
what you might do is say I know there is a vertical asymptote at x equals negative two.
So for a sketch of the graph you can make up some x values. With the vertical asymptote
at negative two you try negative three, negative four, negative five. I like to put these in
order so that is kind of annoying I wrote it like that. On the right hand side of the
asymptote to the right of negative two, maybe you could use the values of negative one,
zero, and one. Plug each of those in, one at a time, into the equation. Like negative
one. Negative one plus seven over negative one plus two. That is six over one, so at
negative one we are at six. So at negative one, my drawing certainly not drawn to scale
very well we would be up here at.... Let's just fix that. Negative one, 1,2,3,4,5, 6.
The graph actually looks like this. I can plug in zero and work it out. Zero plus seven
over zero plus two which is three point five...and so on. I am using some values to the left
of the vertical asymptote and I am using some values to the right of the vertical asymptote
to get an idea of what the graph is doing. For a complete idea, you have to know what
is going on here and why is this curving and flattening out at this value, what appears
to be y equals one. Well, I will take care of that when I cover horizontal asymptotes
in another video. Ultimately we are going to do a few parts and then do one video where
we graph a couple rational functions from start to finish. That video will probably
be shot on Friday. So, that is three examples of how when you have a denominator that can
equal zero you have a vertical asymptote. Well how about and example where we don't
get a vertical asymptote, we get a hole. Y equals x squared minus three x plus two over
x minus one. I can make that denominator equal zero. I cannot use... x minus one cannot equal
zero so x cannot equal one. Something is going to happen at x equals one. Is it going to
be a vertical asymptote again? Well in this case the answer is no! The reason why is going
found through the simplification process on this rational equation. The top trinomial
factors. There are factors of two that add to negative three. I can do that simply because
the leading coefficient is one. The numerator will factor to be x minus two times x minus
one. You can check that. It will multiply together to equal x squared minus three x
plus two. That is over x minus one. Now my x minus one factors cancel out. The simplified
version of this equation is y equals x plus two. That is y equals mx plus b, this is slope
intercept form. This is simply the equation of a line. So I could just graph this by going,
HEY the slope is equal to one over one. That is going to be up one and over one. The y
intercept or b is equal to negative two. That means that is going to, the video is going
to be done in a second.... This line is going to cross the y axis at negative two. You always
do your y intercept first, so that is negative two. It has a slope of one over one, so up
one and over one, up one and over one. And this very complicated looking rational equation
ends up simplifying to be a line. Well, not quite so simply. I have got a very expensive
TI-CAS calculator and even it will show a line without breaks. However, you will have
got to know that originally x could not equal one, so we must back over here where x equals
one and put a nice clear hole at that point. The graph is undefined at one even though
we do not see that vertical asymptote. So any time you have a zero in the denominator
and it factors / cancels out, you will not get a vertical asymptote there you get a hole.
So that is my introduction to vertical asymptotes and holes, let's move on to horizontal ones.
BAM!!!