In Exercises 81–88, a. Find the slant asymptote of the graph of e...

Question

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+x−6)/(x−3)

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the equation of the Slant asom tote of the graph of the given rational function and to graph the given rational function using the seven steps. All right, our given rational function is F of X equals in the numerator, the quantity of X squared minus X minus 20. That's all divided by X minus four. Our answer choices are answer choice. A, a Slant Asymptote, Y equals negative X minus three. And then there is a graph answer choice B the Slant Asom tote Y equals negative X minus three. And then there is a graph answer choice C the Slant Asom tote Y equals X plus three. And then there is a graph and answer choice D the Slant Asymptote Y equals X plus three. And then there is a graph, we will examine these graphs more closely after we complete those seven steps. But right now, we're going to start off by finding the Slant Asymptote. All right, we know that we have a slant Asom tote when the degree of the numerator. In this case, we have a degree of two is one more than the degree of the denominator. In this case, the degree of the denominator is one. So we do have a slant, Asom tote and we find the Slant Asom tote by doing the division. So we're going to take our numerator X squared minus X minus 20. And we're going to divide that by our denominator, which is X minus four. We begin by asking ourselves how many times does X go into X squared? And in other words, X squared divided by X and that's going to leave us with X. And then we take that X written above the same degreed X and we multiply it by R X and our negative four from the denominator X multiplied by X is X squared, X multiplied by negative four is a negative four X. Then we're going to take X squared minus X and subtract the quantity of X squared minus four X or in other words, be sure to subtract both terms, X squared minus X squared is no more X squared and a negative X minus a negative four X or in other words, a negative X plus four, X is a positive three X. Next, we ask ourselves how many times does or excuse me, we can bring down this minus 20. And we ask ourselves how many times does X go into three, X and X goes into three X three times? So we put plus three above the like terms and we could keep going. However, we don't need to know the remainder for the Slant. Asom Tode, we only need this because the Slant Asom tote is equal to this Y equals X plus three. And that's it. That's our slant, Asom tote. So we've done the first part. Yay. All right. Next, we've only got seven steps to graph this so we can get started with those seven steps. I'm going to dispute this down and move our screen a little bit. So the seven steps for graphing, I am going to draw a rough sketch of a coordinate plane. Well, that was pretty rough. We've got a vertical Y axis and a horizontal X axis. They come together at the origin in the middle. The first step of our seven steps to graph is to determine whether there's any symmetry. We find symmetry by substituting N for X A negative X. So we're finding F of negative X. So let's plug negative X in we'll have in the numerator, a negative X squared minus a negative X minus 20 all divided by negative X minus four. And to simplify that negative X squared is positive X squared minus negative X is going to be a plus X still minus all divided by now. A negative X minus four in the denominator. So what signs have changed and what signs have remained the same. Well, the sign of our X squared remains the same. The sign of the negative 20 is the same and the sign of the negative four is the same. However, the sign of our X in the numerator and our X in the denominator have both changed. Now, how do we know if there's symmetry? If we get all of the signs changing or negative F of X, then we have symmetry about the origin. If none of our signs change, if we get what we started with F of X, then we have symmetry about the Y axis. But we had some of our signs change and some of them not change. So we have no symmetry for this particular rational function. So sorry, no symmetry here. But we got step one done. Step two, we are going to find the Y intercepts, we find the Y intercepts by finding F of zero, plugging zero in for X. So let's do that. We'll have zero squared minus zero minus 20 in the numerator all divided by zero minus four. Well, the numerator simplifies to a negative 20. The denominator simplifies to a negative four and negative divided by negative four is a positive five. So we have a Y intercept at zero five. All right. Step two is done two out of seven. We're getting there. Next step is we're going to look for our X intercepts. We find the X intercepts by setting the numerator equal to zero. So we're setting X squared minus X minus equal to zero. I'm going to factor this pretty quickly if you need a refresher on factoring, definitely check out previous lesson video videos, set up two sets of parentheses and we can see that this factors because we need X multiplied by X in our first spots. And then two factors that multiply to be a negative 20 and add up to a negative one that's going to be a negative five and a positive form. So our two factors are X minus five. So we set that equal to zero and X plus four, we set that equal to zero. And now we solve for X. So in the first one, adding five and we get X equals a positive five in the second one, subtracting four for X plus four equals zero. And we get X equals a negative four. So we have two, X intercepts, one of them is at 50 and the other one is at negative 40. All right, we found our X and Y intercepts plugging along our next step. Step four will be to look for our vertical asymptotes. We find our vertical asymptotes by setting the denominator equal to zero. So we have X minus four equals zero. Sol for X, we'll add four to both sides and we get X equals a positive four. So we have a vertical Asymptote at X equals four. That one was quick. Next, we determine our horizontal asom totes step five. We find the horizontal asymptotes by comparing the degrees of our numerator and our denominator. If that sounds familiar, we did already do this and we saw that the degree of our numerator two is one more than the degree of the denominator one, which means that we don't have any horizontal asom totes. But we do have a slant as some tote at Y equals X plus three. OK. Now we do step six, which is to plot some points. And in order to plot points, we want to choose some X values between all of our X intercepts and our vertical asom tote or ASOM totes. So I'm actually going to kind of jump ahead a little bit and work a little bit on step seven, which is to graph so that we can see what X values we need to choose. So I am going to graph what we do know, we know we have Y intercepts at a Y intercept at 05. So from the origin, we're moving neither left nor right for our X value of zero. And we're going up five units for our Y value of five label that 05 for our X intercepts, we have two from the origin for negative 40. We'll go to the left four units and neither up nor down for our Y value of zero. So that's at negative 40. And then our other X intercept is at 50 from the origin moving to the right five units. And then neither up nor down. So there is our 50 X intercept, we have a vertical Asymptote at X equals four. So from the origin, moving to the right four units and then drawing a vertical line for our asom tote E X equals four. And then our horizontal Asymptote, we don't have one, but we do have the slant Asymptote that's in the format of Y equals M X plus B M being the slope. So our slope here is one and B is the Y intercept, which is three. So from the origin, we go up three units and neither left nor right. So we have our slant asy tote intersecting at 03 and then it's going up 1/1, up, 1/1 in the positive direction and then in the negative direction is going left one down, one left, one down one and so on. There is our slant asom tote at Y equals X plus three. So we have graphed all of that information. Now, we can determine what points to plot by making an X Y chart. And we want to choose an X value from each of our intervals. So the first interval is going to be from negative infinity to negative four. The next interval will be from negative four to our positive four. The next interval will be between four and five. That's a smaller interval and the final interval will be between five and positive infinity So I have pre chosen some. You can, you don't have to choose the same ones I do. But I've chosen for the first interval, the X value of negative five for the second interval, the value of zero for the third interval, the X value of 4.5 or nine halves. And for the final interval, the X value of six. And if you want to plug those in, go right ahead, Pause the video, I've already done this. So the Y values corresponding with those X values are for negative five, the Y value is negative 19th. For the X value of zero, the Y value is five for the X value of nine halves, the Y value is negative 17 halves. And for the X value of six, the Y value is five. Now we can plot those points. So from the origin from, for the point negative five, negative 19th, we go to the left five units and then down approximately one unit, negative 19th is about one. For the next one, we already have 05, plotted. Yay, that one's done now for nine halves, negative 17 halves, we go from the origin to the right about 4.5 units and then down about 8.5 units. So that's somewhere down here in the fourth quadrant. And for the 0.65, from the origin, we go to the right six units and up five units which will be approximately here. Now, it's a matter of connecting these points to finish our step seven, which is graphing. So things will typically hug up along our asom totes. So this first line is going to continue down our asom tote of Y equals X plus three toward negative infinity for both the X and the Y values. And then it's going to hit our first point negative five, negative 19th. Then it will hit our X intercept at negative +40. Then it will come up and hit our Y intercept of 05 and then it will come and hug up along our Asom tote at X equals four and then head toward positive infinity for the Y value. There's the first line, the second line is going to hug up along our asom tote at X equals four from negative infinity for the Y value. It's going to hit our point at nine halves, negative 17 halves and will continue and hit our point of 50. Then it will hit our point at 65 and then it will come up and hug along our slant asom tote of Y equals X plus three, continuing toward positive infinity for both the X and the Y values. So there's our graph. Now we just need to find our matching answer choice. So let's evaluate, looking at answer choice. A they have a slant asom tote of Y equals negative X minus three. Well, we have a slant asom tote of a positive X plus three. So we can see looking at the graph that their slope for their slant Asymptote is negative. And so that disqualifies answer choice. A answer choice B again has a slant asom tote of negative X minus three. And they have that negative slope for their slant asom tote. So this one again disqualifies answer choice B All right. Let's move on the screen and we'll check out see and D OK. Four, our answer choice C and answer choice D, they both have the correct slant Asymptote of Y equals X plus three. Now, what's different about them? Well, we can take a look at the vertical asymptotes. We have a vertical Asymptote of X equals positive four. So does answer choice C but answer choice D has a vertical Asymptote of X equals negative four. So that disqualifies answer choice D. And now comparing our graph with answer choice C, it has the same lines following the same pattern of being in the same quadrant and following along the same asom totes. So we will match with answer choice. C well done. We'll catch you on the next one.

In Exercises 81–88, a. Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. f(x)=(x^2+x−6)/(x−3)

Key Concepts

Rational Functions

Slant Asymptotes

Graphing Rational Functions

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