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. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)

Accepted Answer

Hello. In this video, we will go ahead and graph the rational function that is given to us, the function is given as F of X is equal to X squared plus six X minus 55 divided by X squared minus 18 X minus 19. In order to start graphing the rational function, we will first need to find our vertical and horizontal asymptotes. We will start by solving for the vertical asymptotes. Now, the vertical asymptotes are vertical lines that represent the restrictions of the domain or in other words, they are values of the domain that cause the denominator to go to zero. In order to find these values, we will take our denominator set it equal to zero and solve for the values of X that cause the denominator to go to zero. So we will take our denominator which is X squared minus 18 X minus 19 and set it equal to zero and solve. Notice that our denominator is in the form of a factor trinomial. The denominator will factor to give us X minus 19 multiplied by X plus one. We can set both of the factors of X equal to zero and solve. And that will give us X is equal to positive 19 and X is equal to negative one. So this tells us that there are two vertical asymptotes, one located at positive 19 and the other located at negative one. Now that we have the vertical Asymptote, let's go ahead and solve for the horizontal Asymptote. Now, in order to find the horizontal Asymptote, we want to take a look at the leading exponents of the numerator and denominator. The numerator has the highest exponent of two. And the denominator has the highest exponent of two. Since the numerator and denominator have the same value leading exponent the Y vertical Asymptote, I'm sorry, horizontal Asymptote will be equal to the ratio of the leading coefficients of the numerator and denominator. That means our horizontal Asymptote will be Y is equal to one divided by one. And this will simplify to give us Y is equal to one. So we have a horizontal Asymptote located at Y is equal to one. Now that we have the vertical and horizontal asymptotes, we want to go ahead and find the X and Y intercepts. We will start by solving for the X intercepts. Now the X intercepts occur when the function is zero. In order for the function to be zero, we want to find the values of the numerator that cause the num numerator to become zero. So we will take our numerator set it equal to zero and solve that will give us X squared plus six, X minus 55 is equal to zero. Notice that our numerator is in the form of a factor trinomial, our numerator will factor to give us X minus five multiplied by X plus 11. And if we set both of the factors of X equal to zero and solve, we will get X is equal to five and X is equal to negative 11. This means that we have two X intercepts, one located at five comma zero and the other one located at negative 11 comma zero. Now that we have the X intercepts, let's go ahead and solve for the Y intercepts. Now, the Y intercepts occur when the value of X is zero on the graph. So we are going to plug in the value of zero into our function that will give us zero squared plus six, multiplied by zero minus 55 is divided by zero squared minus 18, multiplied by zero minus 19. This will then simplify to give us the value of 55 divided by 19. And that tells us that we have a Y intercept located at zero comma 55 divided by 19. Now that we have the X intercepts and Y intercepts, let's go ahead and try to figure out the behaviors of the graph. Now, the behaviors of the graph can be found by first plotting a number line. What we are going to place on this number line are going to be the vertical asymptotes and the X intercepts. Now, we have the X intercepts at five and negative 11. We also have the vertical asymptotes at 19 and negative one. So we will go ahead and plot those values on our number line that would be negative 11, negative 15 and 19. Now, what we want to do is we want to take testing points between the regions and depending on the value that we get, that will tell us whether the graph will be increasing or decreasing in that region. Let's go ahead and start by taking a testing point to the left of negative 11. A testing point in this region will be negative 12. If we plug in negative 12 into our function, we will get the value 17 divided by 341. Since this is a positive number, the region to the left of negative 11 will be increasing. Next, we will take a testing value between negative 11 and negative one and we will choose negative two. If we plug in negative two into our function, we will get the value negative 63 divided by 21. Since this value is negative, the graph will be decreasing within this region. We will take a testing point between negative one and five which will be zero. And notice that when we plug in zero, we get the value of positive 55 divided by 19. That means that the region is going to be increasing. We will take a testing point between five and 19 which is positive six and plugging this into the function will give us negative 17 divided by 91. Since this number is negative, this region will be decreasing. And if we take a testing point to the right of 19, which is 20 plug it into our function, the output will be 465 divided by 21. This number is positive. So the region to the right of 19 will be increasing. So that was quite a bit of information we just solved for, we solved that the vertical asymptotes are located at 19 and negative one. We have the horizontal asymptotes at Y is equal to one. We have an X intercept at five and 11, negative 11 and we have a Y intercept at zero comma 55 divided by 19. We also figured out the regions of increasing and decreasing for the graph. Now we just need to pick the option that best matches these descriptions and the graph that accurately represents all this information is going to be option D. So the solution to this problem will be D. So I hope this video helps you in understanding of 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. h(x) = (x^2 - 3x - 4)/(x^2 - x -6)

Key Concepts

Vertical Asymptotes

Horizontal and Slant Asymptotes

Graphing Rational Functions

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