In Exercises 57–64, find the vertical asymptotes, if any, the ...

Question

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

Accepted Answer

Welcome back. I am so glad you're here. We are given this rational function F of X equals x minus four in the numerator divided by X plus one. And we get to graph this. We are also to look for horizontal, vertical and slant as um totes while we are graphing. So let's move this down where we've got a bit more room to do some work. Alright. The first step in graphing our rational function is we are going to see if there's any symmetry going on and for looking for symmetry. What we're going to do is we're going to substitute in and negative X for our X values. And we're getting one of three different scenarios. We're going to either get the exact same thing that we started with our original F of X. If that's the case, we'll have symmetry over the y axis. We might get the exact opposite of what we started it started with the negative fx if that So we'll have symmetry about the origin. The third situation is where we don't have exactly what we started with and we don't have the exact opposite and in that case we'll have no symmetry. So let's go ahead and plug in a negative X. So instead of in the numerator X minus four it's negative x minus four. And now in the denominator instead of X plus one, it's negative X plus one. Well that is certainly not what we started with because we have those negatives now and it's not the opposite either because the signs didn't change for these constant terms. So in this case there is no symmetry, which is fine. It's just we know that we won't have any symmetry when we graph it. Okay. Speaking of, let's get a rough sketch of a graph going here. We'll have a vertical Y axis and a horizontal X axis and they meet together at the origin in the middle. Now the next thing we are going to do in our graphing journey is we are going to look for intercepts. So we'll start off looking for y intercepts to find our Y intercepts. We're going to plug in zero for X. So let's do that in the numerator, zero minus four instead of x minus four. And the denominator zero plus one. And so zero minus four is negative four and zero plus one is one. So we have negative 4/1 which is negative four. So when X equals zero, y equals negative four, that is our Y intercept. We can look at our graph and we can plot that. So starting with the origin moving neither to the left or to the right for our X value because that's zero and going down four for our Y value. That's something like this. Zero negative four for our y intercept. Now let's look for our X intercept or intercepts. What we do to find X intercepts is we are going to set our numerator equal to zero. So that is X minus four. We'll set that equal to zero, we can add four to both sides and we get X equals four. Alright, when y equals zero then our X value is four. We can plot this starting at the origin. We're going to move forward to the right for our X value and we move neither up nor down for our Y value at zero. So this is the 00.40. Alright. We have our intercepts plotted. Now. The next step on our graphing journey is we are going to look for assam totes. Alright, so there are three types of ASM totes. We're going to look for vertical, horizontal and slant as mentioned in the problem description. So let's start off with vertical ASM totes for vertical. ASM totes. We are going to set the denominator equal to zero. Our denominator X plus one. Set that equal to zero. We can subtract one from both sides and we get a vertical ASM tote at X equals negative one. Alright, We've got an ASM tote so X equals negative one. We're going to go one to the left of the origin and then we're going to draw our dotted line for our ASM tote meaning that our function cannot come into contact with that or it gets zapped. No, that's not exactly what happens. But it can't go there. So our vertical as X equals negative one. Now horizontal as um totes you can only have one. All right. Do we have a horizontal as well. What we do is we look at the degree of the numerator in this case. That is one and the degree of the denominator in that case it is one. And we are going to compare the relationship of one and one. Well one and one those are equal when the degree of the numerator is equal to the degree of the denominator we recall from previous lessons that we are going to take the coefficient of the numerator. So that's one and divided by the coefficient of the leading coefficient for both of these, For the leading coefficient of the denominator which is also one and our horizontal ASM tote is going to be where Y equals that value and one divided by one equals one. So we have a horizontal ASM tote at Y equals one. Alright, let's look at our graph and we're going to go up one for our Y. Value and then we're going to draw a horizontal line Through why equals one. And there is our horizontal tote. All right. The last type of asem tote is a slant. You have a slant As if the degree of the numerator is one more than the degree of the denominator while they were both one. So the degree of the numerator is not one more than the degree of the denominator. So there is no slant tote. The slant totes are sad but it's really okay. They don't actually really have feelings. Okay. So we've got our ASM totes Now we need to just figure out what is going on with our f of X. We just need to plot some points. Well we know that our only intercepts are at zero negative four and 40. That's the only place that it is going to touch those axes. So we can connect the dots here and we can be confident pretty much in connecting our dots because we know that the function itself is going to approach those two ASM totes so it's going to get as close as it can to those as um totes. And now we also know that that's what's going on to the right of our vertical as um tote of X equals negative one. But what's going on on the left hand side with our function of that vertical as um toad. So what we're going to do is choose a point, choose an X. Value to the left of X equals negative one. And we are going to plug that X value into our original function and then find out the corresponding Y value to see what's going on over there where there's a point. So let's choose negative six. That's to the left of X equals negative one. Are vertical as a toad. So in the numerator will have negative six minus four and in the denominator will have negative six plus one. Well negative six minus four is negative 10 divided by negative six plus one. Well that is negative five. So this simplifies to negative 10 divided by negative five is positive two. Where X equals negative six Y equals positive two. Let's graph that moving from our origin to the left six and then going up to that's something like here. So so it's happening in this area, this quadrant and we know it's going to approach our two assam totes as it continues. So this is roughly what's going on now when we go back up and look for our answer choices, what do we need to look for? Well it'll be a good idea to look for intercepts at zero negative four and 40. And to look for our ASM totes at X equals negative one. And why equals one. Now let's see if I can bring those together. 0 -4. 40 X equals negative one and Y equals one. And bring those back up. And we'll look at our answer choices and try not to get cover them up too much. Okay. All right now we want intercepts at zero negative four and 40. Okay well here's a zero negative four and 40. That's pretty good. B doesn't work. That doesn't have the correct intercepts and zero negative four. But c doesn't have the correct intercepts and for d they have zero negative four but no not the correct intercepts. Okay well is looking pretty good. How about our assam totes does it have an assam tote at X equals negative one, and yes, it does. How about a. An ASM tote at Y equals one? Yes, it does. So answer choice A. Works well done. We'll catch you on the next one.

In Exercises 57–64, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function. f(x) = 2x/(x^2 - 9)

Key Concepts

Vertical Asymptotes

Horizontal Asymptotes

Slant Asymptotes

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