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Let's do a couple more examples

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graphing rational functions.

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So let's say I have y is equal to 2x over x plus 1.

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So the first thing we might want to do is identify our

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horizontal asymptotes, if there are any.

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And I said before, all you have to do is look at the

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highest degree term in the numerator and the denominator.

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The highest degree term here, there's only one term.

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It is 2x.

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And the highest degree term here is x.

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They're both first-degree terms.

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So you can say that as x approaches infinity, y is

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going to be-- as x gets super-large values, these two

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terms are going to dominate.

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This isn't going to matter so much.

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So then our expression, then y is going to be approximately

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equal to 2x over x, which is just equal to 2.

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That actually would also be true as x

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approaches negative infinity.

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So as x gets really large or super-negative, this is going

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to approach 2.

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This term won't matter much.

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So let's graph that horizontal asymptote.

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So it's y is equal to 2.

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Let's graph it.

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So this is our horizontal asymptote right there.

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y is equal to 2, that right there-- let me write it down--

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horizontal asymptote.

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That is what our graph approaches but never quite

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touches as we get to more and more positive values of x or

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more and more negative values of x.

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Now, do we have any vertical asymptotes here?

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Well, sure.

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We have when x is equal to negative 1, this equation or

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this function is undefined.

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So we say y undefined when x is equal to negative 1.

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That's definitely true because when x is equal to negative 1,

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the denominator becomes zero.

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We don't know what 1/0 is.

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It's not defined.

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And this is a vertical asymptote because the x

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doesn't cancel out.

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The x plus 1-- sorry-- doesn't cancel out

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with something else.

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Let me give you a quick example right here.

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Let's say I have the equation y is equal to x plus

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1 over x plus 1.

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In this circumstance, you might say, hey, when x is

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equal to negative 1, my graph is undefined.

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And you would be right because if you put a negative 1 here,

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you get a 0 down here.

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In fact, you'll also get a 0 on top.

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You'll get a 0 over a 0.

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It's undefined.

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But as you can see, if you assume that x does not equal

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negative 1, if you assume that this term and that term are

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not equal to zero, you can divide the numerator and the

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denominator by x plus 1, or you could say, well, that over

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that, if it was anything else over itself, it

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would be equal to 1.

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You would say this would be equal to 1 when x does not

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equal negative 1 or when these terms don't equal zero.

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It equals 0/0, which we don't know what that is, when x is

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equal to negative 1.

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So in this situation, you would not

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have a vertical asymptote.

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So this graph right here, no vertical asymptote.

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And actually, you're probably curious, what does

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this graph look like?

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I'll take a little aside here to draw it for you.

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This graph right here, if I had to graph this right there,

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what this would be is this would be y is equal to 1 for

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all the values except for x is equal to negative 1.

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So in this situation the graph, it would be y is equal

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to 1 everywhere, except for y is equal to negative 1.

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And y is equal to negative 1, it's undefined.

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So we actually have a hole there.

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We actually draw a little circle around there, a little

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hollowed-out circle, so that we don't know what y is when x

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is equal to negative 1.

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So this looks like that right there.

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It looks like that horizontal line.

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No vertical asymptote.

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And that's because this term and that term cancel out when

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they're not equal to zero, when x is not equal to

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negative 1.

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So when your identifying vertical asymptotes-- let me

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clear this out a little bit.

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when you're identifying vertical asymptotes, you want

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to be sure that this expression right here isn't

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canceling out with something in the numerator.

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And in this case, it's not.

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In this case, it did, so you don't

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have a vertical asymptote.

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In this case, you aren't canceling out, so this will

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define a vertical asymptote.

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x is equal to negative 1 is a vertical asymptote for this

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graph right here.

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So x is equal to negative 1-- let me draw the vertical

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asymptote-- will look like that.

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And then to figure out what the graph is doing, we could

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try out a couple of values.

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So what happens when x is equal to 0?

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So when x is equal to 0 we have 2 times 0, which is 0

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over 0 plus 1.

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So it's 0/1, which is 0.

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So the point 0, 0 is on our curve.

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What happens when x is equal to 1?

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We have 2 times 1, which is 2 over 1 plus 1.

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So it's 2/2.

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So it's 1, 1 is also on our curve.

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So that's on our curve right there.

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So we could keep plotting points, but the curve is going

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to look something like this.

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It looks like it's going approach negative infinity as

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it approaches the vertical asymptote from the right.

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So as you go this way it, goes to negative infinity.

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And then it'll approach our horizontal asymptote from the

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negative direction.

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So it's going to look something like that.

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And then, let's see, what happens when x is equal to--

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let me do this in a darker color.

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I'll do it in this red color.

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What happens when x is equal to negative 2?

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We have negative 2 times 2 is negative 4.

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And then we have negative-- so it's going to be negative 4

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over negative 2 plus 1, which is negative 1,

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which is just 4.

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So it's just equal to negative 2, 4.

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So negative 2-- 1, 2, 3, 4.

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Negative 2, 4 is on our line.

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And what about-- well, let's just do one more point.

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What about negative 3?

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So the point negative 3-- on the numerator, we're going to

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get 2 times negative 3 is negative 6 over negative 3

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plus 1, which is negative 2.

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Negative 6 over negative 2 is positive 3.

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So negative 3, 3.

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1, 2, 3.

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1, 2, 3.

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So that's also there.

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So the graph is going to look something like that.

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So as we approach negative infinity, we're going to

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approach our horizontal asymptote from above.

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As we approach negative 1, x is equal to negative 1, we're

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going to pop up to positive infinity.

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So let's verify that once again this is indeed the graph

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of our equation.

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Let's get our graphing calculator out.

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We're going to define y as 2x divided by x plus 1 is equal

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to-- delete all of that out-- and then we want to graph it.

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And there we go.

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It looks just like what we drew.

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And that vertical asymptote, it connected the dots, but we

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know that it's not defined there.

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It just tried to connect the super-positive

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value all the way down.

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Because it's just trying out-- all the graphing calculator's

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doing is actually just making a very detailed table of

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values and then just connecting all the dots.

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So it doesn't know that this is an asymptote, so it

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actually tried to connect the dots.

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But there should be no connection right there.

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Hopefully, you found this example useful.