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

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. r(x) = (x^2 + 4x + 3)/(x + 2)^2

Accepted Answer

Welcome back. I am so glad you're here. We are given this rational function F of X equals in the numerator, X squared divided by the denominator X minus 16. And we are asked to graph this and we need to be sure that we look for horizontal, vertical and slant of some totes if they exist. So let's bring this down where we have more room to work on it. And the first step in graphing our rational function is we are going to look for symmetry. We recall from previous lessons that to look for symmetry. We're going to put in a negative X for our X and we'll get one of three results. If we get the exact same thing that we started with our F of X that we started with, then we're going to have symmetry about the Y axis. If we get the opposite of what we started with the negative F of X, we're going to have symmetry about the origin. If we get neither of those, if it's different from what we started from but not the exact opposite, then we'll have no symmetry. So let's plug in a negative X in the numerator instead of x squared, it's negative X squared in the denominator instead of x minus 16. It's negative X minus 16 in the numerator, negative X squared is x squared in the denominator, negative x minus 16 stays negative x minus 16. And the numerator remains the same as it was originally. They're both x squared. But the denominator that's different. So it is neither exactly what we started with nor the opposite. So in this case there is no symmetry which is fine. But it just means that we don't know we know there's no symmetry when we graphic. So speaking of let's get a graph going, we'll have a vertical Y axis and a horizontal X axis and they meet together the origin in the middle. Now, the next step we're going to take is we are going to look for intercepts we recall from previous lessons that to find the Y intercepts we are going to plug in a zero for X. So let's do that. So in the numerator that will be zero squared in the denominator that will be zero minus 16. 0 square to zero. Zero minus 16 is negative 16. 0, negative 16th is zero. So when X equals zero, Y also equals zero. That means we haven't intercept at the origin. Draw that on the graph. All right. Now, to find any other X intercepts, we are going to set our numerator equal to zero. So that's X squared equals zero while taking the square root of both sides, X on the left and the square root of zero plus or minus zero. That's zero. So there are no other X intercepts. It's just the same one at 00. Okay, we've plotted our intercepts. The next thing we're going to look for is assam totes. There are three types of Assam totes we're going to look for as mentioned before, we're going to look for vertical, horizontal and slant. So for vertical vertical ascent oats we are going to set the denominator equal to zero. So this time X -16 equal to zero. We'll add 16 to both sides and we get a vertical ASM tote at X equals 16. Now we can graph this because I've done this problem in advance. I know that the scale we're going to need to use is very zoomed out. So instead of having 16 be pretty far to the right on the X axis, I'm going to have it look like it's pretty close to the origin. So I'm going to just go to the right a little bit and that can be our X equals 16. And I will label that for our vertical assam tote. Okay, next type of assam tote we're going to look for is horizontal. For our horizontal assam tote we are going to look at the degrees of our numerator and denominator. The degree of the numerator is to the degree of the denominator is one. And so now we're going to look at the relationship between two and one. Well two is greater than one when the degree of the numerator is greater than the degree of the denominator. We recall from previous lessons that there is no horizontal Asam tote, sorry, horizontal assam totes. Now, the last type of as we're going to look for is a slant assam Tote If the degree of the numerator is one more than the degree of the denominator. Then we have a slant as um tote. Well too. Yeah that is one more than one. So yes we do have a slant. As um toad here, vertical. You can have a smiley face to. Alright so we do have a slant as um tote. Now how do we find out what it is? It's going to be the equation of a line and we find that equation by dividing our rational function by our X squared divided by x minus 16. So we're actually going to do that division but we recall from previous lessons that we can use synthetic division here. So we're going to take our C value here. We're going to use 16 and then we're going to use the coefficients and constant term from what's inside the division bracket. So x squared coefficient is one, X is coefficient is zero, there is no X. And the constant term that is equal to zero. There is no constant term. But now we can do the division, bring down the 1 16 times one is 16, 0 plus 16 is 16. 16 times 16 is 256. Zero plus 256 is 256. And now we've got a new equation that is one degree less than what we started with. We started with X squared. So now that's going to be one X or X plus 16 is our constant term plus 256es are a remainder. So 256 over X -16. But this is the equation of the line, ignoring our remainder. The remainder is what tells us that it's going to approach this line. So we have X plus 16. We'll put that into the form of an equation of a line that's why equals M. X plus B. So in this case it will be Y equals what's R. M. Value our slope here is equal to one because there's nothing in front of the X plus B. What's RB here? RB is 16. So our line is Y equals X plus 16. But this M. And the B. Are very helpful for graphing. So now we look at our graph and our Y intercept is up at 16. So starting at the origin we go up and remember we're not making 16. Very high up and then the slope is one so it's up 1/1 up, 1/1. So that's going to do something kind of like this for our slant as um tote and I will label that that is why equals X plus 16. Okay we've got our ASM totes done now we just need to do a little bit more work to figure out what's going on with our function line itself. Well we know where we hit the origin. That we are going to be approaching these assam totes. So it's not going to hit the X axis or the y axis anywhere else and it's going to approach our assam totes. So it looks like this to the left side of our vertical assam. Tote X equals 16. But what's happening to the right side of X equals 16. What's happening over here? We can pick an X. Value and figure out the Point for that associated with it and then see what's going on. So let's graph for x equals 20. Let's see what our y value is. So we'll take f of 20. So we'll have 20 squared in the numerator and minus 16 in the denominator 20 squared is 400 divided by 20 minus 16 is four. So 400 divided by four is 100. So where X equals 20 Why equals 100? Looking at our graph, that's somewhere up here, something like that. So we know we have a point there and we know that our function is going to approach these assam totes. So this is approximately what our function is going to look like on the graph. Now looking for our answer choices. What's the key piece of information we want to look for Because this is very zoomed out. It might be hard to look at the graph and see what our intercepts are but our ASM totes are going to be very helpful in this case. So we know we've got this vertical ASM tote at x equals 60 and we know we have this slant. As um tote at Y equals X plus 16. So let's remember X equals 16 and Y equals X plus 16. Let's go look for X equals 16. Alright. Are looking for a vertical as um toad at X equals 16. And the only vertical ASM tote set two X equals 16 is answer choice B. What is their slant as it is? Why equals X plus 16? That's exactly what we're looking for. So it looks like our answer here is answer choice B. Well done. We'll catch you on the next one.

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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. r(x) = (x^2 + 4x + 3)/(x + 2)^2