Graph each rational function. See Examples 5–9. ƒ(x)=(3x^2+3x-...

Question

Graph each rational function. See Examples 5–9. ƒ(x)=(3x^2+3x-6)/(x^2-x-12)

Accepted Answer

Hello. In this video, we are going to be learning how to draw the graph for the given rational function. The rational function that is given to us is F of X is equal to seven X squared minus seven X minus 210 divided by X squared plus six X minus seven. In order to draw the graph of the rational function, there are a few things that we first need to determine. We'll have to determine the vertical asymptotes and horizontal asymptotes. We will have to determine the X intercepts and Y intercepts and then figure out the behaviors of the graph. So in order to start drawing the function, let's first determine the vertical asymptotes. Now the vertical asymptotes are graphs that represent the restrictions of the domain of the rational function. So in order to find the vertical Asymptote, what we will need to do is we will have to take the denominator of our rational function and set it equal to zero. Once we solve for X, this will give us the restrictions of the domain which will give us our vertical asymptotes. So we are going to take X squared plus six X minus seven and set this equal to zero. What we want to do now is solve for X. Now, our denominator is in the form of a factor trinomial. Our denominator will factor to X plus seven multiplied by X minus one. Using our zero property. We can set the factors of X equal to zero and solve them individually. So we will have X plus seven equal to zero and X minus one equal to zero. On the left hand side, we will have to subtract seven to both sides of the equation to solve for X. And this will give us X is equal to negative seven. And on the right hand side, we will add one to both sides of the equation. And that will leave us with X is equal to positive one. So this tells us that we have two vertical asymptotes. We have one at X is equal to negative seven and we have one at X is equal to positive one. Now that we've determined the vertical asymptotes, let's go ahead and determine the horizontal asymptotes in order to determine the horizontal asymptotes. We want to take a look at the leading exponents of the numerator and denominator. The highest exponent of the numerator is two and the highest exponent of the denominator is two as well. Since the leading exponents on the numerator and denominator are the same, we are going to take the ratio of their coefficients and that will be the horizontal Asymptote. So the ratio of the coefficients is going to be Y is equal to seven divided by one. And this will simplify to leave us with Y is equal to seven. This means that we have a horizontal Asymptote at Y is equal to seven. Now that we have the asymptotes, we have to determine the intercepts. Let's start by determining the X intercept of the function. So how do we find the X intercepts? Well, what we want to do is we want to find the values of X that cause the function to become zero. We know that we cannot plug in zero into the denominator because division by zero is not allowed. But if the numerator was zero, then it will cause the function to become zero. So we are going to take our numerator set it equal to zero and that will give us the values for the X intercepts. So again, our numerator is seven X squared minus seven, X minus 210 is equal to zero. What we could do to simplify this is we could divide every term in this equation by seven. Doing so will allow us to rewrite the equation as X squared minus X minus 30 is equal to zero. The left hand side is in the form of a factor trinomial. And this will factor to leave us with X minus six, multiplied by X plus five and that is equal to zero. And just like before, we are going to set each of the factors of X equal to zero and solve. So we will have X minus six equal to zero. And if we add six to both sides, we will get X is equal to positive six. Furthermore, we'll set X plus five equal to zero. And if we subtract five to both sides of the equation, we will have X is equal to negative five. This tells us that there are going to be two X intercepts one at negative five comma zero and the other at six comma zero. Now that we have the X intercepts, we want to find the Y intercepts. Now the Y intercepts occur when the value of X is zero. So what we're going to do is we're going to plug in zero for every X in our equation that will allow us to rewrite the function as F of zero is equal to seven, multiplied by zero squared minus seven, multiplied by zero minus 210 divided by zero squared plus six, multiplied by zero minus seven. And this will simplify to leave us with positive positive 30. So what this means is that we have a Y intercept at zero comma 30. Finally, we need to determine the behaviors of the graph. Now, the behaviors of the graph are determined by first drawing a number line. What we are going to place on this number line are the values of the vertical asymptotes and the X intercepts and recall that we had a vertical Asymptote at negative seven and positive one at and X intercepts at negative five and positive six. If we label these values on a number line, we will have negative seven, negative five, positive one and positive six. Now, what we want to do is we want to take testing points from each of these regions to determine the behaviors of the graph. Let's go ahead and pick a testing point to the left of negative seven. We can use negative eight. And if we plug in negative eight to F of X, our output is going to be positive 294 divided by nine. Since this is a positive number, this tells us that the region to the left of negative seven will be increasing. And we're going to repeat this process for every other region on the number line. So what's the value that we can pick between negative seven and negative five while we can pick negative six. And if we plug in negative six into our function, the output is going to be negative 84 divided by seven. Since this is a negative number, the graph will be decreasing between negative seven and negative five. A number that we can pick between negative five and one is zero. And we already know that if we plug in zero into the function, our output is going to be positive 30. So that tells us that the region between negative five and one will be increasing. We will pick two to be our testing point between one and six. And this will give us an output of negative 196 divided by nine. Since this is a negative number, this region will be decreasing. And if we pick positive seven to be our testing value to the right of six, the output of the function is going to be 84 divided by 89 which is a positive number. So the region to the right of six will be increasing. Now that we have the regions of increasing and decreasing. We have the asymptotes and we have the X and Y intercepts, we just have to pick the function that matches all this information. Now, just to recap, we have vertical asymptotes at negative seven and positive one. We have a horizontal Asymptote at Y is equal to seven. We have X intercepts at negative five and positive six and we have a Y intercept at positive 30 and we have the regions increasing and decreasing. The graph that matches all of this information is going to be option D. So the solution to this problem is going to be D. So I hope this video helps you in understanding on how to graph this function. And I'll go ahead and see you all in the next video.

Graph each rational function. See Examples 5–9. ƒ(x)=(3x^2+3x-6)/(x^2-x-12)

Key Concepts

Rational Functions

Graphing Techniques

Asymptotes

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