|Start||Now we learn about rational functions. We are not going to do much with rational functions,
but we can do a little bit. So, what is a rational function. A rational function is
some function where you have a P(x), a polynomial on top and a polynomial on the bottom.
For example something like that. Rational functions have four important things that we can find.
|0:43||We can find the vertical asymptotes: we get these from the domain. And a vertical asymptote is a place that you cannot have a value of x plugged into the function, in other words|
|1:05||is where the denominator is zero. So, for example here we can factor the denominator into (x+3) (x+1). So we would know that x cannot equal -3 and x cannot equal -1. So, the domain would be all reals, as long as x does not equal -3 or does not equal -1and|
|1:32||the vertical asymptote would be x=3 and x=1. Let's do another example: y=x^2 -9 over x^2 +8x+7. That factors into (x+7)(x+1), so here x cannot equal -7 and x cannot equal -1. so|
|2:07||that would be the domain. And you have a vertical asymptote at x=-7 and x=-1. A second thing
that we can find is the "end behavior", so these are sometime called horizontal asymptotes.
But there is a difference. A vertical asymptote is a vertical line in the graph that the graph
|2:31||will not cross, so something like that. Horizontal asymptote however isn't the same thing, because you can cross the horizontal asymptote. What we really doing is describing what's called the end behavior. The end behavior just need what happens when x is very large; in the|
|3:02||positive direction or when x is a very negative number, just naturally very large and very small so we would not be hung up on that. So, there is three rules: If you have a polynomial on top and a polynomial on the bottom. So on top you have ax^n plus a bunch of terms and bx^m plus a bunch of terms as many as they are, where n is the highest power on|
|3:33||top and m is the highest power on the bottom. Then there is three possibilities: there is
either n is greater than m, n is equal to m, or n is less than m. That is it. Either
the top is bigger than the bottom, the top and the bottom the degrees are equal, or the
bottom is bigger than the top unless is bigger in degree the highest power. if the top is
bigger than the bottom, then there is no horizontal asymptote or the n behavior is y goes to infinity.
|4:13||If n equals m, then y goes to a over b. If n is less than m, then y goes to zero which would be the x-axis. So what do I mean by "goes to"? If the top is a higher power than|
|4:33||the bottom, then as x gets bigger and bigger , the top become so much bigger than the bottom,
that the bottom starts to became unimportant and the function just gets infinite large.
It goes way up. If the top and the bottom are the same, then the functions becomes to resemble each other because all the smaller terms become less and less important and the function begins to approach a value a over b. And if a bottom is bigger than at some
|5:00||point the bottom is so much bigger that this looks like a small number over a very large number and the function approaches zero. Let's do some examples. If i have y= 3x^4 -7x^2 +1 over 5x^3 +8x +2. Even though this has a square and this only has an x, those become unimportant, so when x is very large y will also become very large and y will go|
|5:35||to infinity. So if you say: is there a number that y approaches? the answer is NO. What is I had y= 3x^4 -7x^2 +1 over 5x^6+8x +2. Now, when x its very large, the bottom becomes so much bigger than the top. One hundredth to the sixth is much bigger than one hundredth|
|6:02||to the fourth. When x is a very big number this is so much bigger than the top. So y would approach zero. And if I had y =3x^4 -7x^2 +1 over 5x^4+8x+2, now when x gets very large y would just approach 3/5. So we sometimes call that horizontal asymptote when the n|
|6:36||behavior is when x is very large y becomes like 3/5. Here when x is very large y becomes like zero, and here when x is very large y goes to infinity. There is a couple other things about rational functions. We can find if x and y intercepts. The value of y when x is zero and the value of x when y is zero. So, let's go back to our example. Well, when|
|7:12||x is zero, you just plug it in: y=0-9 over 0+0+3 , just -9 over 3, or -3. In other words this goes to the point (0,-3): that gives the y-intercepts. To find the value of y when|
|7:34||x is zero is very easy, you just plug in zero for x. What if I want to find the value when
y is zero, well suppose I have x^2 -8x-20 over x^2 -4. Let's not use -4, let's use -9.
|8:01||Sorry! Well, the top factors into (x-10) (x+2) we really don't care what happen in the bottom.
We want to know when y is equal to zero, what is x? this are called the x-intercepts. Well, a fraction is zero when the numerator is zero. So, I just need to know when is the top equal
|8:36||to zero. The top is equal to zero when x is ten or when x is -2. In other words this goes through the point (10, 0) and (-2, 0). So to find where a rational function, the value of a rational function, the value of x in rational function when y is zero, you really have to set the numerator equal to zero, but you need to be careful that the denominator|
|9:00||is not also zero there. We are not going to do any of those. Let's do another example.
Suppose you have y= x^2 +3x-4 over x^2 +25. So, let's look at this. This factors into (x+4)(x-1) over x^2 +25. So here y is zero when x=-4 or when x=1. In other words it goes
|9:39||through the point (-4, 0) and (1, 0). Now, Let's do a complete problem. Suppose I have y= x^2 -6x +8 over x^2 +8x +15 and I want to find the domain, the end behavior where|
|10:16||it crosse the x-axis and where is crosse the y-axes. So the domain: Well I am going to factor the bottom into (x+3)(x+5) and factor the top and the bottom. Ok. The domain is|
|10:57||x cannot equal -3 and x cannot equal -5. The end behavior, well I look at the power of|
|11:08||the top and bottom, the power of the top is x^2 and the power of the bottom is x^2, so when x gets very large, that just looks like x^2 over x^2, y approaches 1. Now, when x equals zero, what is y equal? we just plug in zero, you get (0-4) (0-2) over (0+3) (0+5)|
|11:42||which equals 8 over 15. You can also plug into the original equation and you get 8/15.
So, when x is zero y is 8/15. In other words this goes through (0, 8/15). That is the y-intercepts.
And last, what happens if y is zero? Well when y is zero, I look at my factors: x=4
|12:09||and x+2. In other words this goes through (4, 0) and (2, 0). and those are the x intercepts.