Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College graphing rational functions. Vertical asymptotes occur when there are no common factors in the numerator and denominator. Hence the function is in reduced form and the value has the function being undefined. Simple terms. What makes the denominator go to 0? Horizontal asymptotes or slant asymptotes? The line that the function approaches as X goes to Infinity or X goes to negative Infinity? If degree in the denominator is greater than the degree in the numerator, then the horizontal asymptote is going to be y = 0. If the degree in the denominator equals the degree in the numerator, then the horizontal asymptote is going to be Y equal the leading coefficient of the numerator divided by the leading coefficient of the denominator. If the degree in the denominator is less than degree in numerator by exactly 1°, then a slant asymptote occurs and is found by polynomial division where the remainder is ignored because the remainder as we get really really really far into a big X value or really really far into a negative big X value. A constant over really really big X is going to go to 0 examples of graphing rational functions. We want to find the X intercept, Y intercept, vertical asymptote, and horizontal asymptote or oblique asymptote and put it all together. So the X intercept is always found by setting Y equal to 0. If we look at this, what's really going to always always happen with that denominator? Well, when we cross multiply, the denominator is always going to go away. So the X intercept is really just going to be found by setting the numerator equal to 0. In this case, X is going to equal 0. the Y intercept we're going to put zero in for our X, and in this case we're going to get 0 / 3, which is really zero. So the X intercept and Y intercept is the same. The vertical asymptote occurs from what makes the denominator go to 0, the horizontal asymptote. We look at the degree of the numerator and the degree of the denominator. In this case, the degree is one understood one on that plain old X and an understood one on the quantity X + 3. So the horizontal asymptote is going to be the leading coefficient in the numerator over the leading coefficient of the denominator, or Y equal 5. If we put all this together on a graph, we have our 00, we have a vertical asymptote at X equal -3, and we have a horizontal at Y equal 5. Now we actually know how our graph's going to look just based on that. We have this intercept here. We know that we can't ever cross the X axis anywhere except at an X intercept. So if we look to the left of our horizontal or of our vertical asymptote, sorry of X equal -3, can I ever cross the X intercept to the left of X equal -3? The answer is no. So I know that my graph has to get close to the horizontal asymptote when we're at negative Infinity and get close to the vertical asymptote as my X's get closer to -3. This X equal -3 has an odd multiplicity. If it's got an odd multiplicity, if the Y values are on positive on one side, then on the other side of that asymptote, my Y values are going to be negative When I come up to this intercept, what's the multiplicity there? Well, it's odd again, so we're going to go through it. IE the Y values on one side were negative, so the Y values on the other side have to be the opposite, in this case positive. And when I get out to Infinity, I need to get close to that horizontal asymptote. Now, it's important to understand we can cross a horizontal asymptote, but we can never cross a vertical asymptote. So that's what our graph is going to look like. If we look at another example, we're going to have X intercepts wherever that numerator is 0. So in this case -1 zero Y intercept, we're going to put zero in for our X. So 0 + 1 is 1. So -1 in the top 0 + 2, two squared is 4. The vertical asymptotes occur when the denominator can't be 0. So in this case, X equal -2 is a vertical. The horizontal. The degree on top is less than the degree on bottom. So we're going to have a horizontal at y = 0, so negative 100, negative 1/4, X equal -2, and y = 0. That's enough to tell us how this graph's going to look like. When I'm to the right of X = 0. Am I ever going to have an X intercept? No, because I don't ever cross the X axis. So I know I've got to be close to that horizontal asymptote and I've got to go through the Y intercept. I'm going to come up to the X intercept. That X intercept had what kind of multiplicity? Well, it was an understood 1, so it's odd. If it's odd, the Y values on one side are negative, so the Y values on the other side are going to have to be the opposite or positive. Now X equal -2 what kind of multiplicity? Here it's even, So the Y values are going to do the same thing. If the Y value is positive on one side of that vertical asymptote, then the Y value needs to be positive on the other. I don't have any X intercepts going less than -2, so I'm not going to cross the X axis, but I'm going to get closer to y = 0, and this one we need to factor first. So we're going to get X + 1 on top X + 3 X -2, so our X intercepts when the numerator equals 0, our Y intercept, we put zero in for X and we're going to get a negative 1/6. Our vertical asymptotes when is the denominator equal to 0, so X equal -3 and X equal 2 and the horizontal, the degree on bottom is bigger, so it's going to be y = 0. Putting all this together, we should be able to come up with what our graph is going to look like. X equal -3 and X equal 2 y = 0. So the fact that we have this point here, we don't have any X intercepts in between zero and two. So I know that I'm going to come up from my vertical to get to my Y intercept, I've got to go to my X intercept next. That X intercept had an odd multiplicity because that X + 1 is understood to be to the one power, so odd multiplicity. So it's going to go through the Y values on one side are going to be negative, the Y values on the other are positive. This intercept here at X equal -3 also an understood one or an odd multiplicity. So if the Y values are positive on one side, the Y values have to be negative on the other. Over here, the X equal to understood 1, so it's an odd multiplicity. So if we've got negative Y values on one side, the other side has to be the opposite or positive. We don't have any X intercepts greater than two, so I'm going to get closer and closer and closer to the horizontal asymptote. This next one we're going to factor to start. So we're going to have X + 3 X -2 / X + 2. The X intercepts are going to be when the numerator equals zero. Negative 3020 Y intercepts X is 0. What's our Y in this case -3 Vertical asymptotes occur when the denominator can't be 0. Now this time we don't have a horizontal asymptote, we have a slant asymptote, or sometimes called an oblique asymptote. And that's because the degree on top is exactly one more than the degree on bottom. So what we're going to do is we're going to do synthetic division or polynomial long division. Either way, if I do my synthetic division, I'm going to get X -, 1 -, 4 / X + 2. Now what happens when we're way, way, way out to Infinity or negative Infinity to this -4 / X + 2? Well, 4 / a really, really, really, really big number really goes to 0 as X goes to positive and negative Infinity. So our slant asymptote or our oblique asymptote is Y equal X -, 1. If we put all this on a graph, Y equal X -, 1 X equal -2 zero -3 is a point -3 zeros a point and two zeros a point. So we know we have to be close to the slant asymptote at negative Infinity. I've got to come to my X intercept. Was that an odd or an even multiplicity? Well, this one came from right here, so it's an odd multiplicity. Hence we go through and we're going to come out to that asymptote. This asymptote came from this X + 2 being to an understood first power, which is odd. So if the Y values on one side are negative, they have to do the opposite on the other side of that asymptote. The next thing that we're going to is the next X intercept, which is -3 it's got an understood 1, so it's odd. So it's going to pass through and get closer to this asymptote as we get out to negative Infinity. One more example, this one's going to be a little more complex. Same steps, X intercept. What makes the numerator go to 0? So in this case -1 zero negative 40. the Y intercept is if I put zero in for X. And in this case there aren't any because we'd end up with a zero in the denominator and we can't divide by zero. The vertical asymptotes are going to occur at X = 0, X equal -2, and X equal 3. The horizontal, if we look at the total degrees 2453 plus 2 is 5 S, 5 and 5 S They're the same, so it's going to be the leading coefficient, which in this case is just one over 1. So we have a negative 10 a -4 zero, X = 0, X equal -2, X equal 3 and Y equal 1. Lots of information there on the graph. So when we graph this can the graph, if I'm looking at to the right of X equal 3, could it go like this? And the answer is going to be no because we don't have an X intercept out that direction. So we know that the graph has got to come like so getting close to both those asymptotes. Now, this line here at X equal 3 came from right here, and its multiplicity is even. So if the Y values on one side are positive, the Y values on the other side have to also be positive. Do I have any intercepts in here in between these two vertical asymptotes? The answer's no. So I can't cross the X axis. So that tells me I've got to come back up and get closer to that next vertical asymptote. Well, the next vertical asymptote came from this plain old X. That one's got an odd multiplicity. So if the Y values on one side are positive, the Y values on the other side are going to have to be the opposite or negative. This time we do have an X intercept. So when I go up to this X intercept, that came from -1 So that came from this term here, which is an even multiplicity. So the Y value on one side is negative. If it's an even multiplicity, the Y values have to do the same thing. So now it's got to be negative, IE the concept of a bounce. The next thing we're going to look at is this next asymptote, X equal -2, which came from this term. X equal -2 is an even multiplicity. So if the Y's are negative on one side, the Y's have to be negative on the other side. The next thing we're going to do is go up to the intercept. This intercept came from X equal -4 up here in the numerator. It's an odd multiplicity. So if the Y's are negative on one side, the Y's have to be positive on the other because it's odd, and then it's going to get close to that horizontal asymptote as we go out to negative Infinity. Thank you and have a wonderful day.