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- Let's do a couple more examples
- graphing rational functions.
- So let's say I have y is equal to 2x over x plus 1.
- So the first thing we might want to do is identify our
- horizontal asymptotes, if there are any.
- And I said before, all you have to do is look at the
- highest degree term in the numerator and the denominator.
- The highest degree term here, there's only one term.
- It is 2x.
- And the highest degree term here is x.
- They're both first-degree terms.
- So you can say that as x approaches infinity, y is
- going to be-- as x gets super-large values, these two
- terms are going to dominate.
- This isn't going to matter so much.
- So then our expression, then y is going to be approximately
- equal to 2x over x, which is just equal to 2.
- That actually would also be true as x
- approaches negative infinity.
- So as x gets really large or super-negative, this is going
- to approach 2.
- This term won't matter much.
- So let's graph that horizontal asymptote.
- So it's y is equal to 2.
- Let's graph it.
- So this is our horizontal asymptote right there.
- y is equal to 2, that right there-- let me write it down--
- horizontal asymptote.
- That is what our graph approaches but never quite
- touches as we get to more and more positive values of x or
- more and more negative values of x.
- Now, do we have any vertical asymptotes here?
- Well, sure.
- We have when x is equal to negative 1, this equation or
- this function is undefined.
- So we say y undefined when x is equal to negative 1.
- That's definitely true because when x is equal to negative 1,
- the denominator becomes zero.
- We don't know what 1/0 is.
- It's not defined.
- And this is a vertical asymptote because the x
- doesn't cancel out.
- The x plus 1-- sorry-- doesn't cancel out
- with something else.
- Let me give you a quick example right here.
- Let's say I have the equation y is equal to x plus
- 1 over x plus 1.
- In this circumstance, you might say, hey, when x is
- equal to negative 1, my graph is undefined.
- And you would be right because if you put a negative 1 here,
- you get a 0 down here.
- In fact, you'll also get a 0 on top.
- You'll get a 0 over a 0.
- It's undefined.
- But as you can see, if you assume that x does not equal
- negative 1, if you assume that this term and that term are
- not equal to zero, you can divide the numerator and the
- denominator by x plus 1, or you could say, well, that over
- that, if it was anything else over itself, it
- would be equal to 1.
- You would say this would be equal to 1 when x does not
- equal negative 1 or when these terms don't equal zero.
- It equals 0/0, which we don't know what that is, when x is
- equal to negative 1.
- So in this situation, you would not
- have a vertical asymptote.
- So this graph right here, no vertical asymptote.
- And actually, you're probably curious, what does
- this graph look like?
- I'll take a little aside here to draw it for you.
- This graph right here, if I had to graph this right there,
- what this would be is this would be y is equal to 1 for
- all the values except for x is equal to negative 1.
- So in this situation the graph, it would be y is equal
- to 1 everywhere, except for y is equal to negative 1.
- And y is equal to negative 1, it's undefined.
- So we actually have a hole there.
- We actually draw a little circle around there, a little
- hollowed-out circle, so that we don't know what y is when x
- is equal to negative 1.
- So this looks like that right there.
- It looks like that horizontal line.
- No vertical asymptote.
- And that's because this term and that term cancel out when
- they're not equal to zero, when x is not equal to
- negative 1.
- So when your identifying vertical asymptotes-- let me
- clear this out a little bit.
- when you're identifying vertical asymptotes, you want
- to be sure that this expression right here isn't
- canceling out with something in the numerator.
- And in this case, it's not.
- In this case, it did, so you don't
- have a vertical asymptote.
- In this case, you aren't canceling out, so this will
- define a vertical asymptote.
- x is equal to negative 1 is a vertical asymptote for this
- graph right here.
- So x is equal to negative 1-- let me draw the vertical
- asymptote-- will look like that.
- And then to figure out what the graph is doing, we could
- try out a couple of values.
- So what happens when x is equal to 0?
- So when x is equal to 0 we have 2 times 0, which is 0
- over 0 plus 1.
- So it's 0/1, which is 0.
- So the point 0, 0 is on our curve.
- What happens when x is equal to 1?
- We have 2 times 1, which is 2 over 1 plus 1.
- So it's 2/2.
- So it's 1, 1 is also on our curve.
- So that's on our curve right there.
- So we could keep plotting points, but the curve is going
- to look something like this.
- It looks like it's going approach negative infinity as
- it approaches the vertical asymptote from the right.
- So as you go this way it, goes to negative infinity.
- And then it'll approach our horizontal asymptote from the
- negative direction.
- So it's going to look something like that.
- And then, let's see, what happens when x is equal to--
- let me do this in a darker color.
- I'll do it in this red color.
- What happens when x is equal to negative 2?
- We have negative 2 times 2 is negative 4.
- And then we have negative-- so it's going to be negative 4
- over negative 2 plus 1, which is negative 1,
- which is just 4.
- So it's just equal to negative 2, 4.
- So negative 2-- 1, 2, 3, 4.
- Negative 2, 4 is on our line.
- And what about-- well, let's just do one more point.
- What about negative 3?
- So the point negative 3-- on the numerator, we're going to
- get 2 times negative 3 is negative 6 over negative 3
- plus 1, which is negative 2.
- Negative 6 over negative 2 is positive 3.
- So negative 3, 3.
- 1, 2, 3.
- 1, 2, 3.
- So that's also there.
- So the graph is going to look something like that.
- So as we approach negative infinity, we're going to
- approach our horizontal asymptote from above.
- As we approach negative 1, x is equal to negative 1, we're
- going to pop up to positive infinity.
- So let's verify that once again this is indeed the graph
- of our equation.
- Let's get our graphing calculator out.
- We're going to define y as 2x divided by x plus 1 is equal
- to-- delete all of that out-- and then we want to graph it.
- And there we go.
- It looks just like what we drew.
- And that vertical asymptote, it connected the dots, but we
- know that it's not defined there.
- It just tried to connect the super-positive
- value all the way down.
- Because it's just trying out-- all the graphing calculator's
- doing is actually just making a very detailed table of
- values and then just connecting all the dots.
- So it doesn't know that this is an asymptote, so it
- actually tried to connect the dots.
- But there should be no connection right there.
- Hopefully, you found this example useful.