 In this video we provide the solution to question number 15 for practice exam number three for math 1050 We're given a rational function f of x equals x squared times x minus six all over x plus four Times x minus three squared We have to graph this function, but it also asks us to list some informations like what's the x intercepts of the graph That's the first thing it mentions the x intercepts are going to come from the Rational functions numerator what makes the numerator go to zero and then it's in reduced simplified form There's no common factors here, so it's already simplified. We're going to get intercepts at zero when this guy goes to zero and Six when this factor goes to zero we needed we need to indicate their multiplicities here so notice that zero has an even multiplicity and Six has an odd multiplicity. This tells us that at zero we will touch the x-axis, but not cross at six We will cross the x-axis To find the y intercept There's only the one that happens that when x equals zero so we plug zero into the formula there You're going to end up with zero squared times negative six All over four times negative three squared. This just turns out to be zero So let's label what we discovered so far so zero zero was an x intercept and a y intercept and then we had x equals six Going forward here. So one two three four five six Like so. All right. No big deal there Next Let's consider the asymptotes. So let's look for the vertical asymptotes. These are going to be the things That make the denominator go to zero again when you're in reduced form If you had something that cancelled out that could give you a remove point there isn't one in this example though So we have an x plus four we have an x minus three squared So we have vertical asymptotes at x equals negative four and at positive three again It asks us to list the multiplicities negative four has an odd multiplicity of one three has a even multiplicity of two So at negative four we are going to cross infinity, but at positive three we are going to touch infinity So I want to add these to my graph here So we have a vertical asymptote at negative four add that in there And I'll just add a little label here. This is x equals negative four and then we also have one at positive three Do your best to keep these things straight Usually like to do a dash line to emphasize. This is a vertical asymptote This is a place we want to avoid x equals three in this situation And then the last thing it wants to also consider it wants all asymptotes So is that our bleak asymptote is there whores us until what's the in behavior of this thing? So notice that the in behavior is going the in behavior here We're just going to look at the leading terms. So you have this x squared Times x over x times x squared again That is a week nor everything except for the leading terms This looks like x cubed over x cube, which really just comes becomes a one This is going to be our horizontal asymptote And so let's add that horizontal asymptote to the graph now There's no scale on the y-axis just on the x-axis. That's okay. You see the little tick marks right here We didn't do that on the y-axis and that's because the scale of the y-axis doesn't really matter so much for these pictures It doesn't have to be a perfect one a perfect scale vertically speaking Mostly it's we were trying to pin tension. How does it how does the function behave near its intercepts and near its asymptotes? That's what we're looking for do label these things of course especially since there's no scale in play here So this is y equals one. Okay, and so with that now in play. We're ready to start drawing our picture What's going on here? So we have for example our intercept here at six So we have x equals six right here. What's gonna happen? Well, if we're at this intercept, we have to somewhat we have to we have to either go up or we have to go down Right, those are only two options. Well, do we go up towards the asymptote the horizontal asymptote? Are we go down towards whatever? Well, if you go down, you're gonna have to at some point turn around Right, and there's no x-intercept so you can't cross or you have to go off the screen and come back to the side But there's no vertical asymptote. So those options are not available to us So it seems that we have to go up from x equals six there So go up and bend towards your Horizontal asymptote right there We could we probably should be checking is it do we cross or horizontal asymptote? We do something like this. Is there ever a point where the function equals one? You can check that out real quickly There is no such location This one it doesn't cross it's horizontal asymptote that doesn't mean that's always the case but in this one We don't Since at six we cross the x-axis. We have to come back down from the other side, right? Now we can't go back up again because well, we don't have an x-intercept. We have to go off towards infinity Meaning we have to go closer and closer to our vertical Ascent of there at neg or a positive three now at positive three Remember we are going to touch infinity. So we're gonna come back from the same side We started with draw a little arrows right there Then we're gonna come towards x equals zero x equals zero we touch the x-axis So we're gonna come back down like so again. There's no x-intercepts to go up from so therefore We're gonna have to go down towards our Vertical asymptote at negative four we cross infinity so we come up from the other side Like so and again, there's no more exit. There's no more x-intercepts or any more asymptotes So we're gonna have to start approaching our horizontal asymptote giving us something like this All right, so let me draw that last part again a little bit better a little bit more smooth But in general a picture is gonna look something like the following. It's not perfect It's a little crude. You see my finger wiggling a little bit as I'm drawing this not a big deal Even though it's crude. This is exactly the type of picture. We want we should label things of course So we have the intercept zero zero. We have the intercept zero six zero We already labeled the asymptotes x equals negative four x equals three and y equals one And so that was everything it asked us to do we listed the multiplicities of the x-intercepts and the red asymptotes So we're good. This then finishes question number 15