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

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. g(x) = (2x - 4)/(x + 3)

Accepted Answer

Hello. In this video, we are going to be graphing the rational function by finding the horizontal vertical and slant asymptotes. If there are any, the function that is given to us is F of X is equal to 11 X minus 121 divided by X plus 17. So in order to graph this rational function, there's a few things that we're first going to need to find. First, we're going to need to find the vertical asymptotes. Now, in order to find the vertical asymptotes, we need to find the restrictions of the graph. In order to find the restrictions, we want to find the values of X that cause the denominator to become zero. In order to do this, we will take what is in our denominator, which in this case is X plus 17 and solve for the values of X, this will be the values of X that cause the denominator to become zero. Thus resulting in undefined values. In order to solve for X, we will subtract 17 to both sides of the equation that will give us X is equal to negative 17. And this tells us that we have a vertical Asymptote located at X is equal to negative 17. Next, we will want to find the horizontal asymptotes. In order to find the horizontal asymptotes. We want to take a look at the highest degrees of the numerator and denominator. The highest degree of the numerator is one and the highest degree of the denominator is one as well. Since the degrees of both the numerator and denominator are the same, the horizontal Asymptote will equal to the ratio of the leading terms. In both the numerator and denominator. The ratio in this case will be Y is equal to 11 divided by one. And once we simplify, that will tell us that our horizontal Asymptote will be located at Y is equal to 11. Now that we have the horizontal Asymptote, let's go ahead and try to find the X intercepts the X intercepts occur when the function goes to zero on the X axis. So we know that we don't want the values of zero to be in the denominator. But what we want to know is that if we get the values in the numerator to be zero, then that will cause the entire function to be zero. So in order to find the X intercepts, we're going to take our numerator which is 11 X minus 121 and set it equal to zero. What we want to do now is solve for X notice that both of the terms in this equation are multiples of 11. So if we divide the entirety of the equation by 11, we can reduce the equation to be X minus 11 is equal to zero. Next, we will go ahead and add 11 to both sides of this equation and we will get X is equal to 11. That tells us that there will be an X intercept at the 0.11 0. Now that we have the X intercept, let's go ahead and try to find the Y intercept. Now the Y intercept occurs when the output of the function is zero. So in order to find the Y intercept, we're going to plug in zero into our function that is going to give us F of zero is equal to 11 multiplied by zero minus 121 divided by zero plus 17. Once we simplify this, we will have the value of negative 121 divided by 17. And that tells us that there will be a Y intercept at the 0.0 comma negative 121 divided by 17. Finally, we want to go ahead and try to find the behaviors of the graph. Now, in order to find the behaviors of the graph, the first thing we want to do is to create a number line. What we are going to do with this number line is we are going to less our X intercepts and our X are vertical asymptotes So we know that we have a vertical Asymptote at X is equal to negative 17 and we have an X intercept at positive 11. So in our behaviors, we are going to mark negative 17 and positive 11. And what we want to do is we want to pick values that we can plug into our function and determine whether we get a positive or negative number. If the number we get is positive, that tells us that the function will be increasing in that region. And if we get a negative value that tells us that the function will be decreasing in that region, we will start by t taking the testing point to the left of negative 17. And the testing point that we can try will be negative 18. If we plug in negative 18 into our function, the output is going to be 319. Since the output is a positive number, the graph will be increasing to the left of negative 17. Now let's go ahead and take a testing point between negative 17 and positive 11. A testing point that we can pick is zero. But notice that when we plug in zero into the function, we get the Y intercept, which is a negative number. So that tells us that the graph will be decreasing between negative 17 and positive 11. Finally, we will take a testing point to the right of positive 11 and we can choose 12. If we plug in 12 into our function, the output will be 11 divided by 29. Since this is a positive number, the graph will be increasing to the right of positive 11. So that was quite a bit of stuff. We just found, we know that the vertical Asymptote is located at negative 17. We have a horizontal Asymptote at positive 11. We have an X intercept at 11 comma zero. We have a Y intercept at zero, comma negative 121 divided by 17. And we know that the graph will be increasing to the left of negative 17 to the right of positive 11 and decreasing in between negative 17 and 11. With that being said, the graph that matches this information is going to be option B and this is going to be the solution to this problem. So I hope this video helps you in understanding how to graph a rational function. And I'll go ahead and see you all in the next video.

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. g(x) = (2x - 4)/(x + 3)

Key Concepts

Vertical Asymptotes

Horizontal Asymptotes

Slant Asymptotes

Watch next