Oblique Asymptotes

Graph the rational functio...

Question

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = x² / (x − 1)

Accepted Answer

Hello, in this video, we are going to be graphing the given rational function and its asymptotes. The function given to us is Y is equal to X2 divided by X minus 3. Now, let's go ahead and start this problem by finding the asymptotes of the rational function. First, if we want to solve for the vertical asymptotes, what we want to do is we want to find the values of X that cause the denominator to be zero. Division by zero causes an undefined value. Therefore, if the denominator is zero, we get a vertical asymptote. In order to solve this value, we will take our given denominator, X minus 3, and solve it for 0. And when we solve for X, we get X is equal to 3. Which is going to be the vertical asymptote of this graph. Now, are there any asymptotes to the graph? Well, one thing to notice is that the degree of the numerator is higher than the degree of the denominator by exactly one value. What this means is that we are going to have a slant asymptote. In order to determine the graph of the slant asymptote, we need to go ahead and perform polynomial long division. We are going to take our numerator and divide it by the denominator. That is going to be X minus 3, divides X2. Now we want to do is we want to go ahead and perform the polynomial long division. So, we will go ahead and start by multiplying X minus 3 by X. That is going to give us X2 minus 3 X. And when we subtract this value, we have to distribute a negative sign across X2 minus 3X, allowing us to switch the signs. That is going to cancel out the X2 terms, leaving us with 3 X. Now, a value that we can multiply to X minus 3 to get us as close as possible to 3 X will be 3. If we multiply X minus 3 by 3, that will give us 3X minus 9. And again, once we distribute a negative sign across 3X minus 9, we get -3 X plus 9. This cancel out 3X, leaving us with the remainder of 9. Now for the slant asymptote, we can disregard the remainder. We only want the quotient that remains, and that is going to be X + 3. So, the equation of our slant asymptote. is going to be Y is equal to X + 3. Now that we have the equations of the asymptotes, what about the horizontal asymptote? Well, in this case here, because the graph has a slant asymptote, that means that the horizontal asymptote will not exist. So, we don't have any horizontal asymptote. Now, let's go ahead and solve for the intercepts, starting with the X intercept. In order to find the X intercept, we want to find the values of X that cause the numerator of the rational function to be zero. In order to do this, we will take our numerator X2, set it equal to 0, and solve. When we take the square root of both sides of this equation, we get X is equal to zero. What this means is that we have a X intercept at the origin, and since we have the X intercept at the origin, that also tells us that the Y intercept will be at the origin as well. So we have a point located at 0.0, which is the origin. Now let's go ahead and determine the behaviors of the graph around the vertical asymptote. In order to determine the behaviors of the graph, we are going to take the left and right hand limits of the graph, around the vertical asymptote. Again, the rational function given to us is Y is equal to X2 divided by X minus 3. Let's go ahead and check the limit as X approaches the vertical asymptote located at 3 from the left. So we have the limit, as X approaches 3 from the left of X2 divided by X minus 3. Now, since there is no way to further simplify this function, we are going to use the numerical method. So, let's just go ahead and check values of X that slowly approach 3 from the left hand side. We can go ahead and plug in the values of 2.9. 2.99 and 2.999. When we plug in the value of 2.9, we get the value of 84.1. When we plug in 2.99, we get negative 894.01, and when we plug in the value of 2.999, we get negative 8,994.001. Notice that as we get closer to the value of 3, the graph is going to continue decreasing. And since we have an infinitely decreasing value, we can assume that the behavior as the limit approaches 3 from the left is going to be negative infinity. Now, let's go ahead and check the right hand side of the vertical asymptope. We will take the limit as X approaches 3 from the right of X2 divided by X minus 3. And again, we are going to have to use the numerical method to analyze this behavior. We will pick values that are to the right of 3, which are going to be 3.1, 3.01, and 3.001 to plug into the graph. When we plug in 3.1, we get the Y value 96.1. When we plug in 3.01, we get 906.01, and when we plug in 3.001, we get 9,0006.001. As the values are approaching 3 from the right hand side of the vertical asymptote, the outputs are growing infinitely, and that means that the behavior to the right of the vertical asymptote is going to be positive infinity. Now that we've determined the behaviors and we've determined. The vertical asymptote and the slant asymptote. Let's go ahead and plot the graph. So, we will start by plotting the asy coats. The vertical asymptote is located at 3. So this is going to be X is equal to 3, which is going to be the vertical asymptote. Now we need to, now we need to plot the slant asymptote. Now, the slant ascenttote is Y is equal to X + 3, which is going to be the equation of line with the slope 1, and it starts at 3 on the Y axis. Next, we have, or let's go ahead and plot this slant asymptote. Now we have. An intercept at the origin. And finally, we just determined that as this graph is approaching the vertical asympto from the left, it is going to decrease to positive infinity. And if we draw the left hand behavior with respect to the asymptote, this is going to be the shape of the graph on the left hand side of the vertical asymptote. And on the right hand side, we just determined that the behavior of the graph approaches positive infinity when it approaches the right of the vertical asymptote. So, if we draw the graph with respect to the asymptotes, this is going to be the behavior of the graph. And this is going to be the rough sketch of the function Y is equal to X2 divided by X minus 3. So I hope this video helps you in understanding on how to approach this problem, and I will go ahead and see you all in the next video.

Oblique Asymptotes

Graph the rational functions in Exercises 103–108. Include the graphs and equations of the asymptotes.

y = x² / (x − 1)

Key Concepts

Rational Functions