Graph each rational function. See Examples 5–9.

Question

ƒ (x) = x / (4 - x^{2})

Accepted Answer

Hello. Today we are going to be drawing the graph for the given rational function. The rational function that is given to us is F of X is equal to seven X divided by 64 minus X squared. In order to graph a rational function, there's a few things that we're going to need to find. We're going to have to determine our vertical and horizontal asymptotes, determine the X and Y intercepts and determine the behaviors of the graph. So let's go ahead and start by finding the vertical asymptotes. The vertical asymptotes are asymptotes that represent the restrictions of the domain of our given function. Since our function is a rational function, in order to refine the restrictions of the domain, we are going to have to take our denominator and find the values that make it zero. So we are going to take 64 minus X squared, set it equal to zero. And whatever values that we get for X are going to be the restrictions of the domain which will give us our vertical asymptotes. In order to start this process, we will first add X squared to both sides of this equation. This will give us 64 is equal to X squared. In order to simplify X squared, we will have to take the square root of both sides of this equation. This will give us X and the square root of 64 will be eight. But we need to be careful because keep in mind whenever we take a square root of a number, we always get a positive and negative version of that number. This means that there are two numbers that give us restrictions on the domain. They're going to be positive eight and negative eight. So we have two vertical asymptotes. We have a vertical Asymptote at X is equal to positive eight. And we have a vertical Asymptote at X is equal to negative eight. So now that we've determined the vertical asymptotes, let's go ahead and find the horizontal asymptotes. In order to find the horizontal asymptotes, we will need to take a look at the leading exponents of the numerator and denominator looking at our function, the leading exponent of the numerator is one and the highest exponent of the denominator is two. Since the exponent of the numerator is smaller than the exponent of the denominator, this means that we can automatically assign the horizontal Asymptote to be Y is equal to zero. So that is going to be our horizontal Asymptote. Next, we are going to determine the X intercepts. Now the X intercepts occur when the equation or the function becomes zero, we understand that we can't have zero in the denominator because that will give us an undefined value. But if we have zero in the numerator, that will cause the function to go to zero. So in order to find the X intercepts, we are going to take our numerator which is seven X and set it equal to zero. Now, what we are going to do is we are going to solve for X. In order to solve for X, we will just have to divide both sides of this equation by seven. And that will leave us with X is equal to zero. Since we have X is equal to zero, this tells us that we have an X intercept occurring at the origin. Furthermore, because the X intercept is occurring at the origin, that also tells us that the Y intercept will occur at the origin as well. Finally, we will have to determine the behaviors of the graph. So how do we determine the behaviors of the graph? Well, in order to do this, we are going to have to create a number line. What we are going to put on this number line are the values of the vertical asymptotes and the values of the X intercepts. So we determined that we have two vertical asymptotes, one at positive 81 at negative eight. And we understand that the X intercept is occurring at the origin. So we are going to label negative eight zero and positive eight on our behaviors. And what we are going to do is we are going to take testing values from each of these regions to plug into our equation. And that will tell us if the graph is going to be increasing or decreasing in that area, we will go ahead and start by taking a testing point to the left of negative eight. And we can choose negative nine. If we were to plug negative nine into our equation, the output will be 63 divided by 17. Since this is a positive number, this tells us that the graph will be increasing to the left of negative eight. A testing point that we can pick between negative eight and zero is positive negative one, negative one is between negative eight and zero. If we were to plug in this value into our equation, this will give us negative seven divided by 63. Since this is a negative number, this tells us that the graph will be decreasing between the region of negative eight and zero. If we pick a positive one to be the testing point between zero and eight and plug it into our function. This will give us the value of positive seven divided by 63. Since this is positive, the graph will be increasing in this region. And if we pick a testing point to the right of positive eight, which is positive nine and plug it into our function. We will get the value of negative 63 divided by 17. And since this is negative, our graph will be decreasing to the right of positive eight. Now that we have the behaviors, the asymptotes and the intercepts, we just have to pick the graph that matches these descriptions. The option that matches these descriptions is going to be option A, we have the vertical asymptotes located at negative eight and positive eight. We have the intercepts occurring at the origin 00. And we know that the graph will be increasing to the left of negative eight and to the left of positive eight and decreasing to the right of negative eight and to the right of positive eight. With that being said, the solution to this problem is going to be a. So I hope this video helps you in understanding on how to graph a rational function. And I'll go ahead and see you all in the next video.

Graph each rational function. See Examples 5–9.
$ƒ (x) = x / (4 - x^{2})$

Key Concepts

Rational Functions

Domain and Vertical Asymptotes

Graphing Rational Functions and Asymptotes

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