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^2−1)/x

Accepted Answer

Welcome back. I am so glad you're here. We are asked to determine the equation of the Slant Asymptote of the graph of the given rational function. And to graph the given rational function following the seven steps, our given rational function is F of X equals in the numerator two X squared minus three. That's all divided by X answer. Choice A has a Slant asom tote of Y equals two X. And then there is a graph answer choice B has a Slant asom tote of Y equals two X. And then there is a graph answer choice C has a Slant asom tote of Y equals negative two X. And then there is a graph and answer choice D has a slant asom tote of Y equals negative two X. And then there is a graph, we will look at the differences between those graphs later on. But for now, let's start off with finding the Slant Asom tote. And then we will work on those seven steps of graphing. So for the Slant Asom tote, we look at the degrees of our numerator and our denominator, the degree of the numerator is two, the degree of the denominator is one. When the degree of the numerator is one more than the denominator, then we have a slant Asymptote. And in this case, yes, two is one more than one. So we do have a slant Asymptote. Now, how do we figure out what it is? We are going to take our numerator two X squared and we want to enumerate each degree of X. So it's going to be two X squared plus zero X and then minus three. And we're dividing that by our denominator, which is X. So first we take two X squared divided by X. That's going to leave us with two X. We put that above the like term that zero X and so two X multiplied by X gives us that two X squared that we had, we subtract two X squared minus two X squared and that's no more X squared. We bring down the zero X. Now we ask ourselves what is zero X divided by X? Well, and that's just zero. So we have a plus zero here and we could continue with the division. However, for the Slant Asymptote, we only need this part of it. We don't need the remainder. So we can set our Slant Asymptote is Y equals and we have two X and we don't need to write the plus zero. This is in the format of Y equals M X plus B where M is the slope and B is the Y intercept. So our plus zero, that's our Y intercept. B is our Y intercept, which is zero and our slope is a positive two. Our M is a positive too. So there's our slant as tote Yay, we did the first part. Now we only have seven more steps to go in order to graph this. So let's move to a different part of our screen where there's a little more room. And now the first step, I'll draw a rough sketch of a coordinate plane with a vertical Y axis, a horizontal X axis. They come together at the origin in the middle. The first step is we are going to see if there's any symmetry for symmetry, we are finding F of negative X and then seeing what happens. So plugging a negative X in for of our XS in the numerator will have two multiplied by a negative X squared minus three all divided by negative X. Simplifying in the numerator negative X squared is just X squared. So we still have two X squared minus three in the numerator all divided by negative X. Well, we can factor out that negative from the negative X in the denominator. So this is a negative for the whole fraction of two X squared minus three in the numerator divided by X in the denominator. And what we have here is the negative F of X. And when we have the negative F of X as our results we do have symmetry and it's symmetry about the origin. So this time we do have symmetry about the origin. That is step one completed. I apologize for pauses while I clear that space up. All right. Step two is we are going to find any Y intercepts. We find the Y intercepts by finding F of zero or plugging zero in for X. So in the numerator, we'll have two multiplied by zero squared minus three, all divided by zero. Well, we could simplify the numerator but note that our denominator here is zero and we can't have that, that's undefined. So there are no Y intercepts for this particular problem. Next step three, we're going to look for X intercepts, we do that by setting the numerator equal to zero. So we'll have this time two X squared minus three equals zero. We can add three to both sides. We have two X squared equals a positive three, divide both sides by two. We have X squared equals three halves, take the square rut of both sides. And on the left, we just have X and on the right, we have plus or minus the square root of three halves. So we have two X intercepts. One is at a negative square of three halves zero. And the second one is it the positive square root of three halves zero. Those are two X intercepts. And for later on, when we need that three halves is approximately 1.22. So we have a plus or minus 1.22. So we've got our X intercepts. Step number four will be to find the vertical asom totes. We find the vertical asymptotes by setting the denominator equal to zero. Well, this one's quick, the denominator is X so X equals zero. That's our vertical asym tope. Great. Now, first step five, looking for horizontal asymptotes, we find horizontal asom toads by comparing the numerators uh or excuse me, comparing the degrees of our numerators and our denominators. And if that sounds familiar, yeah, that's what we started off doing. Our numerator is one degree higher than our denominator. And when that happens, there are no horizontal asom tode. However, we do have the slant asom tote which is at Y equals to X step number six is to plot points. We need to plot points in between for all of our intervals. So we have intervals between our X intercepts and our vertical asymptotes. So in order to do that, I am going to actually work on step seven for a minute, which is to graph. So I'm going to plug in all of the information we know from steps one through six. So first we have our X intercepts at negative square root of three halves zero. So from the origin, we're going to the left approximately 1.22 units for our point negative square root three halves zero. And then we'll go from the origin to the right, approximately 1.22 units for our point positive three halves zero. And then our vertical asom tote is along the Y axis at X equals zero. And then we do have the slant as to, we don't need that for plotting points, but we'll get it on the graph. And right now anyway, so we recall from earlier that our Y intercept is at the origin 00. And we have a slant or excuse me, we have a slope of positive two. So we'll go up to over one, up to over one. And from the origin, we can also go down to, to the left one, down to, to the left one. And so we can draw a slant as tote mine's not the straightest line, but that's just approximate and that's our slain asom tote of Y equals to X. OK. So now figuring out where to plot those points. Our first interval is from negative infinity to negative three halves. Our second interval is from negative three, er excuse me, not negative three halves, but negative square root of three halves. Our second interval is from the negative square rut of three halves, 20. The next interval is from zero to positive square root of three halves. And the final interval is from positive square root of three halves to positive infinity. So we have four intervals and I am choosing for those intervals, the X values of negative two, negative one, positive one and positive two. And plugging those in to the original function. You, if you want to do that, go ahead and pause the video. But I've already calculated this for X equals negative two. The Y value is a negative five halves for X equals negative one. The Y value is a positive one for X equals positive one. The Y value is a negative one and for X equals a positive two, the Y value is a positive five halves. And that all makes sense because we do have origin about the symmetry or symmetry about the origin. Now, we can plot those points from the origin. We have the point negative two negative five halves. So we'll go to the left two units and we will go down about 2.5 units. So that's approximately here. And for our point negative 11 from the origin, we'll go to the left one unit and then we will go up one unit approximately here. For a 0.1 negative one from the origin, we will go to the right one unit and down one unit approximately here. And for our final point from the origin, going to the right two units for our X value of two and then going up 2.5 units for our Y value of five halves. So approximately here. And the last thing that we do is we just connect all of these points and draw our lines. Now, often these will bump up against our Asom totes. So the way this will behave is from negative infinity, we have going along our slant asom tode of Y equals two X. On the left side of that Asom tote from negative infinity. For both the X and Y values, it's coming up along that slant asom tote, passing through our point negative two negative five halves, coming up through our X intercept of negative square root three halves, then coming up through our point negative 11 and then heading up toward positive infinity for the Y value along our vertical asom tode of X equals zero. So there's our first line. And now for the second one, we will be to the right of our vertical Asymptote of X equals zero. We can start from negative infinity for the Y value. And then it will come up along the vertical Asymptote pass through the point of one negative one, then through our point for our X intercept of square three halves zero and then through our point of 25 halves and then come up along our slant asom tote of Y equals to X heading toward positive infinity for both the X and Y values. And there's our graph. Now we just need to see which answer choice matches with this. So let's look, we bring this to a different part of the screen. We'll look at answer traces a and B first. All right. For answer choice A, they have a slant Asymptote of Y equals two X. That's also what we have looking at the graph. There is the vertical Asymptote along the Y axis, the horizontal or excuse me, the slant Asymptote of Y equals two X and the lines are through the correct parts of the quadrants and on the correct sides of the asymptotes. So this one looks like a great match. Let's compare it to our other answer choices and make sure we haven't missed anything. Answer choice. B has the slant asom tote of two X. But the lines are drawn on the other sides. When we plotted our points, we had the first line to the left of both the slant Asymptote and the vertical asom tote and answer choice. B's graph has it to the left of the vertical Asymptote but to the right of the slant asom tote. So we can rule out answer choice. B let's look at answer traces C and D. All right. For answer choices, C and D, they both have a slant asom tote of negative two X and that's an, a negative slope. That's not what we have. So we can eliminate both answer choices C and D and we can be rest assured that we are correct with our answer choice. A 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^2−1)/x

Key Concepts

Slant Asymptote

Polynomial Long Division

Graphing Rational Functions

Watch next