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 − 4) / (x − 1)

Accepted Answer

Hello. In this video, we are going to be graphing the following rational function and its asymptotes. We are told that Y is equal to X2 minus 16 divided by X minus 2. So, let's go ahead and start by solving for the vertical asymptote. In order to find the vertical asymptote of this graph, we want to find the value of X that causes the denominator to be 0. Division by zero causes an undefined value, therefore producing a vertical asymptote on the graph. We will take the denominator X minus 2, set it equal to 0, and solve. And solving for X is going to give us the value X is equal to 2, which is going to be the vertical asymptote of this graph. Now, what about a horizontal or oblique slash slant asymptote? Notice that the degree of the numerator is higher than the degree of the denominator by one. Since this is the case, this indicates that we have a slant asymptote. So, in order to determine the slant asymptote, we want to go ahead and perform polynomial long division. We will take the numerator X2 minus 16, and divide it by the denominator X minus 2. So, a number that we can multiply to X minus 2, that gets us as close as possible to X2, is going to be the value positive X. X minus 2 multiplied by X, will give us X2 minus 2 X. And when we subtract this quantity, we have to distribute a negative sign across X2 minus 2 X. That is going to give us negative X2 + 2 X. The escort terms will cancel out, leaving us with 2 X minus 16. Now, a number that we can multiply X minus 2 to get as close as possible to 2 X is going to be 2. X minus 2 multiplied by 2, will leave us with 2 X minus 4. And again, when we distribute a negative sign across 2X minus 4, that will leave us with -2 X plus 4. The two x terms cancel L. Leaving us with -12 as the remainder. Now, for the slant asymptote, we can go ahead and ignore the remainder. We only want to focus on the quotient. Since the polynomial long division gave us X + 2, that means the equation to the slant asymptote is going to be Y is equal to X + 2. Furthermore, because we have a slant asymptote, there is not going to be a horizontal asymptote. A graph does not have a horizontal asymptote and a sled asymptote at the same time. So, now let's go ahead and solve for intercepts. We'll start by solving for the X intercept. In order to find the X intercept, we want to find the values of X that cause the numerator to be zero. That is going to give us X2 minus 16 equal to 0. In order to solve for X, we will add 16 to both sides of the equation, leaving us with X2 equal to 16. And if we take the square root of both sides of the equation, we get X is equal to positive and negative 4. This gives us two X intercepts. We have the 0 4.0 and the point 4.0. Now let's go ahead and solve for the Y intercept. In order to solve for the Y intercept, we want to plug in the value of 0 into our equation. That is going to give us Y is equal to 02 minus 16 divided by 0 minus 2. This will simplified to be 16 divided by 2, which is going to give us 8. So we have a Y intercepted located at the point 0.8. Now we need to check the behaviors of the graph around the vertical asymptote located at 2. In order to do this, we can take those one-sided limits of both sides of the vertical asymptote. Starting with the limit as X approaches to from the right, we will take the limit of the given equation X2 minus 16. Divided by X minus 2. Since there is no further way we can simplify the given limit, we will go ahead and use the numerical approach to evaluate this limit. So we are going to pick values that are slightly to the right of 2. Some values that we can plug in are going to be 2.1. 2.01 and 2.001. When we plug in 2.1 into the graph, we get the value negative 115.9. When we plug in -2.01, we get -1,195.99, and when we plug in 2.001, we get negative 11,995.999. Notice that as we get closer to 2 on the right hand side, the value of the graph decreases infinitely, and that means that the limit is going to be negative infinity on the right hand side of the of the given limit. Now let's go ahead and solve for the left hand limit around the vertical asympto. That is going to be the limit as X approaches to from the left of X2 minus 16 divided by X minus 2. And again, we are going to use the numerical method to solve this limit. We are going to pick values to the left of two, which are going to be 1.9, 1.99, and 1.999. When we plug in 1.9 into the equation, we get the value 123.9. When we plug in 1.99, we get 1,203.99. And when we plug in 1.999, we get 12,0003.999. Notice that as we get closer to on the left hand side, our value of the graph increases infinitely, and that means that the graph is going to increase to positive infinity on the left hand side of the vertical asymptote. Now let's go ahead and plot all of the information we solved for. So, starting with the vertical asymptote, we know the vertical asymptote exists and X is equal to 2. We also know that we have a slant asymptote at X Y is equal to X + 2. That is going to be the equation of a line with a slope of 1 starting at 2 on the Y axis. So this is going to be the rough graph of the slant asymptote. Furthermore, we have X intercepts located at -4 and 4. And finally, we have a Y intercept located at 8. We know that to the left of the vertical asymptote. The graph is going to increase infinitely, and to the right of the vertical asymptote, the graph is going to decrease infinitely. So, if we draw the behaviors of the graph with respect to the asymptotes, this is going to be the rough shape of the given graph. And this is going to be the final shape of Y is equal to X2 minus 16 divided by X minus 2. So I hope this video helps you in understanding on how to plot this type of graph, 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 − 4) / (x − 1)

Key Concepts

Rational Functions