 So in this video I want to demonstrate how does one graph a rational function a general rational function here There's a couple steps you would want to go through and you don't necessarily have to go through it in this step In this process right here, but this does summarize all the things we want it We want here to graph a rational function the first thing that we consider is the domain of the function the domain is what? Makes the denominator go to zero so the denominator, which is a polynomial You said it equal to zero you solve that and any number that makes the denominator go to zero We throw out of the we throw that out of the domain Okay, and so the things that make the denominator go to zero we'll come back to that in just a second Like I said, you don't have to go in this order So once you determine the domain these are things these are points to avoid on the graph It's also important to find out the intercepts of the graph The first the the wire steps is usually pretty easy the wire set just comes about by plugging in x equals zero into the function Assuming zero did not make the denominator go to zero This you'll have a y-intercept and then that's a point you would plot on the graph I'd also want to find the x-intercepts The x-intercepts are going to be places that make the denominator the numerator excuse me go to zero in this case Now another thing I want to mention before we go on here is once you figure out the domain What makes the denominator go to zero after that after you determine that then I would want you to simplify the fraction That is put it into lowest terms only after you determine the denominator That's that's why step one needs to be first on this thing put this thing into lowest terms when you found out the denominator Okay, because x-intercepts will only be those points After you put into lowest terms Right, so we need to figure out what makes the numerator go to zero in that situation If you want to you can test for symmetry This is often listed in college outer textbooks. It can be useful, but it's not like super amazing So you can check to see whether the functions odd or even because if your graph has symmetry You basically just have to graph the right hand side with respect to the y-axis and then by reflection or rotation Then you can get the other side as well. But like I said, it's not a super helpful thing You'll find out this information anyways We do want to graph the vertical asymptotes the vertical asymptotes will be those points Which make the denominator go to zero when you're in lowest terms So we need to find out the we need to figure out the vertical asymptotes of the graph Now we do need to know about the Multiplicities of these asymptotes. I'm kind of jumping ahead to step six right here We do need to determine the multiplicities of the x-intercepts and the vertical asymptotes because even means you'll touch the x-axis or cross our touch infinity on Multiplicities means you'll cross the x-axis or cross infinity Depends whether you're looking at x-intercepts or vertical asymptotes there now, of course if there was a point that In the unsimplified form, right that made the top go to zero and zero at the same time Like if you get the zero over zero you have to kind of simplify this thing out Right and that's this can really lead to some remove points something We might talk about in a future example a little bit more detail here But in simplified form if something makes the denominator go to zero it was a vertical asymptote If it makes the numerator go to zero it was an x-intercept We also care about the in behavior of the graph this might lead to things like horizontal asymptotes Which occur when the graph is bottom-heavy or if it's balanced you can also get an oblique asymptote When the top is one degree higher than the bottom you get an oblique asymptote But even if it's top-heavy your graph will asymptotically approach a polynomial So knowing that asymptotic polynomial could be of use So once you know the intercepts the asymptotes and their multiplicities Then I would start plotting these things and I don't mean like a mad scientist We're plotting domination of the world or anything like that I mean we're gonna put the points on the graph the intercepts the asymptotes and then connect the dots Once we have that information there So let's look at an example here. Let's graph the function r of x equals x minus 1 over x squared minus 4 to find the Denominator or to find the domain. We're gonna factor the denominator. It's a difference of squares We get x minus 2 and x plus 2 Which if you set if you set the denominator equals zero by the zero product property We see that the domain will be everything except for two and negative two So those will be outside the domain of the function Let's look for some intercepts. The wider set is when you plug in x equals zero so you get negative one over negative four, which gives you one fourth If you want to figure out the x intercepts you just have to set the numerator equal to zero In which case you're going to get x equals one in that situation And so this is information that then I would start plotting right so we would plug in the point the y-intercept right here Y equals one fourth. We plug in the x-intercept of one. Those are things to consider there All right The next step in our list was test for symmetry like I said, this isn't gonna be crazy helpful But if you want to test for symmetry you're gonna take r of negative x you replace e to the x is with a negative sign in The numerator you're gonna get negative x minus 1 Right and the denominator you're gonna negative x square actually becomes a positive x square there x squared minus 4 So the denominator the negative sign disappeared by the numerator it didn't disappear. So Not all the negative signs disappeared. So it's not gonna become r of x It's not even but you also can't just factor a negative sign out the whole thing There's no symmetry here. Like I said, I'm not gonna play a lot of significance on that Coming over to the vertical asymptotes the vertical asymptotes This this function right here is in lowest terms. There was no common factors on top and bottom So the things that make the geometry go to zero are going to be 2 and negative 2 For which then we get vertical asymptotes at 2 and negative 2 notice that in this example that the x-intercept X minus 1 showed up once so that's an odd multiplicity and For each of the vertical asymptotes those are odd multiplicities as well So everything in this thing has a multiplicity of 1 In terms of horizontal asymptotes, you'll notice that this function is bottom-heavy the denominator is quadratic The numerator is linear. So as x approaches Positive or negative infinity we see that y will approach zero you have a horizontal asymptote Now do we approach it from above or below? That's a little bit trickier question And which case then just imagine what's gonna happen here as x gets big big big big big right as x approaches infinity here As x approaches infinity you're gonna get something y is gonna be approaching something like infinity minus 1 over Infinity squared minus 4 all of this other stuff is peanuts compared to infinity here This thing is gonna look like basically 1 over infinity when you're done, which is 0. That's a positive 0 So we're gonna approach 0 from above, but as x approaches negative infinity things are a little bit different here You're gonna get y is now approaching negative infinity minus 1 over Negative infinity squared minus 4 again. This stuff is peanuts compared to infinity But this is now gonna look like notice negative infinity squared is actually going to be a positive So the denominator is still gonna be positive But the numerator now is actually gonna be negative negative infinity This is gonna look like negative 1 over infinity, which actually looks like 0 from below And so we're gonna see that as we graph this thing although this this analysis I just did a moment ago is not something you actually need to do it's sufficient for us to know that We're gonna approach the horizontal asymptote of y equals 0 So if I plug this information into My graph I see the following so as x approaches infinity We see that we're gonna approach the x-axis from above as we approach negative infinity will perhaps the x-axis from below that part We infer we've inferred just a moment ago. What happens near the x the x intercept of 1 Well, when you're close to the x intercept of 1 our Function will basically look like the following r of x R of x will be approximately equal to when x is approximately equal to 1 Basically, you're gonna plug 1 into everywhere in the function except for that one factor that made it go to 0 So you're gonna get x-1 on top on below. You're gonna get 1 squared minus minus 4 So this is gonna look like x minus 1 over Negative 3 so this looks like negative one-third x minus 1 So this is gonna look essentially like a line with a negative one-third slope. So it's gonna be very shallow And it's gonna be it's kind of negative so it's gonna look something like that that's where this picture comes from right here When x is close to 2 we're gonna plug in 2 everywhere in the function except for that one part that makes it go to 0 So this is gonna look like r of x R of x is gonna look like 2 minus 1 over x minus 2 we don't plug in we don't plug in there because we get a 0 we're gonna get 2 plus 2 and So that simplifies to be 1 over 4 times x minus 2 so this is gonna look like a reciprocal function with us with a Slope of 1 fourth basically so as x approaches Infinite as x approaches 2 from the right It's gonna come up and as x approaches 2 from the left It's gonna point down because that's what these reciprocal functions look like and so we get that behavior right there Okay, and then if we did this process again Erasing that If we did this process again as x is close to negative 2 in the situation r of x Will be approximately the same thing as negative 2 minus 1 over negative 2 minus 2 And then I'm gonna leave the x alone on the next part because that would make it go to 0 the denominator there So you get negative 3 over negative 4 times x plus 2 So that becomes 3 over 4 times x plus 2 so this will look like a standard a Standard reciprocal function so again of odd degree so it's gonna go up on the right It's gonna go down on the left so that's where all the information came from right there But I want to demonstrate for you that this analysis about oh am I approaching and I'm going up or down left or right That type of stuff is really actually not necessary when you do this But before we do that, let me just kind of connect the dots we have right here If we connect all of this information together We connect all of this together We get the picture that we see right here the final product So Graphing this thing we have all this redundant information that comes together and computes this for us I want to try this problem one more time and Look at the information we have so I'm gonna graph the My x-axis right there. Whoops my x-axis and y-axis and so I'm proving to this to you that I'm gonna do this without any Technology whatsoever and so what did we notice here? What do we know about the graph? So we had an x intercept at 1 right we had a vertical asymptote at 2 So you get something like that. Well, maybe make it look a little bit more vertical And then we had another vertical asymptote at negative 2 so we're gonna mention those right there So we have it's good to label your graph here x equals 2 was a vertical asymptote X equals negative 2 was a vertical asymptote There was a the x-axis itself was a horizontal axis until y equals 0 And then we had x equals 1 was an x intercept and then we also have that y intercept of 1 4th so y equals 1 4th and it turns out that with this information and knowing the Multiplicities we actually have all the information we need right here So let's start with the y intercept right here, right? It has to go off towards basically the x intercept somehow or another and just like a polynomial functions There's there's a limit on how many times you can turn this thing around, right? And so we're gonna draw this picture with minimal turns We also want it to be Continuous except from we hit a hole in the domain and it needs to be a smooth as well no sharp corners, right? Those don't happen on rational functions. So basically what happens is we have to start approaching x equals 1 So we're gonna have to come down towards x this x intercept right here And since it had odd multiplicity that tells us we have to cross the x-axis and as we cross it We're gonna come back down. We're gonna have to go down now as we get close to this vertical asymptote We either have to go down towards infinity or we have to turn around and go off towards positive infinity That's the only possibilities here, but wait a second if we're gonna go off towards positive infinity That would mean we have to cross the x-axis again, but there's no other x intercepts The only x intercept was at x equals 1 So because there's no x intercept allowed there. We're gonna have to continue on down towards x equals or we got to go up down towards negative infinity and So here we had this crossing we had a cross at the x-axis But as we get close to the x inner or the vertical asymptote again We're gonna cross infinity which tells us we have to wrap around from the other side. All right And so that means we're gonna have to cross infinity like so then we have to get closer and closer towards our horizontal Acetote which we're gonna have to approach it from above. How do I know we have to approach it from above? Well, if we didn't approach it from above that's because we'd have to maybe if we approached it from below We'd have to do something like this, but wait a second if we're approaching it from below I mean, they're having another x intercept which there isn't one right and so then We put that information together since there's no other x intercepts. We're gonna have to approach At y equals 0 from above so that gives you the right hand side of the graph if we had symmetry We could reflect to get the other side. We don't know that but hey when we're at the y intercept We're either gonna have to approach infinity from for our vertical asymptote or we have to approach negative infinity Those are the only options when we get close to it vertical asymptote But way to get again, there's no x intercept there, right? We can't cross down to go towards negative infinity Therefore the graph is gonna have to go up towards positive infinity and since the since negative two had odd multiplicity We're gonna cross infinity come up from the other side So we get something like this and then the same thing is again Do we approach our horizontal asymptote from above from below or do we approach it from above, right? Well, it can't be the second option because again that would require an x intercept We don't have and so this actually gives us the picture that is necessary here And now compare the one I drew by hand versus the one the computer produced and you can see that it's actually pretty good, right? We had our vertical asymptotes, which we found right here We had our x intercepts and the behavior near the asymptotes near the intercepts was correct Using multiplicities we can graph this entirely here and that's typically how I'd like to graph these rational functions Identify the intercepts identify the asymptotes and then using the test like a starting point like I usually like use the y intercept Using a starting point you can then navigate through the intercepts and through the asymptotes using the multiplicities Do I cross or do I touch at each of those locations? And so that's going to conclude for us a lecture 31 and lecture 32 We're going to use some more examples of this because this can get really complicated very quickly And so we'll do some more examples of this in our next lecture Stay tuned