Vertical Asymptotes How mathbff
Hi guys! I'm Nancy. and I'm going to show you how to find the vertical asymptotes of a rational function. If you don't know what I mean by rational function just ignore that, you're probably in the right place. Actually all this stuff is a lot easier than it sounds. So don't worry. It's gonna be fine. Let me show you.
OK, so say you need to find the vertical asymptote or vertical asymptotes, if there's more than one. First of all. what is a vertical asymptoteé It's basically an invisible vertical line that your graph approaches and gets really really close to but never actually touches. Never quite reaches. So how do you find the vertical asymptoteé Well, if you have a rational function, the fraction form type,
which is the most common kind for this, you can use the same 3 steps always to find the vertical asymptote. OK. So here are the 3 steps you can always use to find the vertical asymptotes of a rational function. The first is to factor the top and bottom. If you can and all that you can. I'll show you what I mean.
The second is to cancel any common factors from the top and bottom to simplify. And the last step is to take the bottom, the denominator and set it equal to zero to find the vertical asymptotes. So let's try it for this function. The first step is to factor the numerator and denominator. if you can. And all that you can.
So you can't factor the top. in any way, but the bottom you can factor. It's a quadratic. And if you want more help on how to factor quadratics I have a tutorial for that. So you can check that out. But basically the idea is you want to break this into. 2 separate factors that start with x.
So x plus a number times x plus another number. OK, and what you want are 2 numbers here. 2 numbers here that. multiply to 6 and add to 5. And those 2 numbers would be 2 and 3. Since 2 times 3 is 6 and 2 plus 3 is 5. So in here we have 2 and 3. And that's all that you can factor for this rational function.
How to Find the Horizontal Asymptote mathbff
Hi guys! I'm Nancy. and I'm going to show you super easy ways to find the horizontal asymptote for any rational expression that you're given. There are basically 3 cases and if you know those 3, then you're set. So to do this you're going to need to know what the degree of a polynomial is.
So for instance if your polynomial is: x^2 + 3x + 4 The degree is the highest power of x that appears in your polynomial. So in this case x^2 is the highest power of x and the degree is 2. The degree is 2, so it's that power. OK, so this is a trick to know the horizontal asymptote.
It's a rule that you can use. So the first case: What if your degree on top is less than your degree on the bottomé So in this case, the degree on top is 1 for the x. and 2 for the x^2. If your degree on top is less than the degree on the bottom then your horizontal asymptote HA
will always be y = 0. which is the xaxis. So that would be the answer. Whenever the degree on top is less than the degree in the denominator your horizontal asymptote is y = 0. Second case: If your degree on top is the same as your degree on the bottom
and in this example that's x^2 on top, x^2 on bottom so the degree is 2 on top and 2 on the bottom then your horizontal asymptote you will get by dividing the leading term on top by the leading term on bottom. So x^2 divided by 3x^2
and simplify and this leaves you with 1 over 3, since that was just a 1x^2. So your horizontal asymptote is y = 13 So if the degrees are equal on top and bottom you're going to end up with a horizontal asymptote that is: y = a number. OK, third case: