Graphing Simple Rational Functions

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Question

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = −3/(x − 3)

Accepted Answer

Hello. In this video, we are going to be sketching the graph of the following rational function and include the graphs of the asymptotes. So, we are given Y is equal to -4 divided by X + 4. In order to start this problem, let's go ahead and determine the function of the vertical asymptote. In order to find the function of the vertical asymptote, we want to solve for the value of X that causes the denominator to be zero. We are going to take the denominator X + 4, set it equal to 0, and solve. And when we solve for X, that is going to give us X's equal to -4, which is going to be the function of the vertical asymptote. Now, what about any horizontal asymptotes or slant asymptotes? If we take a look at the function, notice that the degree of the numerator is lower than the degree of the denominator. What this means is that there is no slant asymptote, but there could be a horizontal asymptote. In order to determine the horizontal asymptote, we are going to take the limit as X approaches infinity of the function -4 divided by X + 4. When we direct substitute this limit, we get -4 divided by infinity plus 4. If you add anything to infinity, you will be left with infinity. So we have -4 divided by infinity, and any number divided by infinity will give us the value of 0. So we have a horizontal asymptote located at Y is equal to 0. Now that we've determined the asymptotes, let's go ahead and determine any intercepts, starting with the X intercept. Now, in order to find the X intercept, we want to find any values that cause any values of X that cause the numerator to be zero. But since there are no values of X that can cause the numerator to be zero, there is not going to be any X intercepts. Now, what about the Y intercepts? In order to find Y intercepts, we want to plug in the value of 0 into our function. That is going to give us Y is equal to -4 divided by 0 + 4. That is going to simplify to be -4 divided by 4, which will further simplify to be -1. So we have a Y intercept located at 0-1. Now what we want to do is we want to determine the behaviors around the vertical asymptote. In order to determine these behaviors, we are going to take the left and right hand limits of the given function. We will go ahead and start by taking the limit as X approaches -4 from the left of the function -4 divided by X + 4. Since we can't simplify the function any further, we are going to use the numerical method where we take values slightly to the left of 4, plug it into the function, and determine a behavior in the outputs. Some values we can choose are going to be negative 4.1, 4.01, and -4.001. When we plug in -4.1 into the function, we get the value of 40. Plugging in 4.01 will give us 400, and plugging in -4.001 will give us 4000. Notice that as we are getting closer to -4 on the left hand side, the values are increasing infinitely, and that means that the behavior of the graph is going to approach positive infinity to the left of the vertical asymptote. We are going to repeat this process, but when we approach the right hand side of the vertical asymptote, when the limit as X approaches -4 from the right. Now, again, we are going to use the numerical method to determine this behavior. Some values we can pick to the right of -4 are going to be -3.9, -3.09, and -3.009. So, when we plug in these values, we will get the output of -3.9 to be 40. When we negative 40 apologies, and when we plug in -3.09, we get -400 and -3.009 will give us -4000. Notice that as we are getting closer to the value of -4 from the right hand side, the values of the graph are decreasing infinitely, and that means the behavior of the graph is going to approach negative infinity to the right hand side of -4. Now what we want to do is you want to go ahead and plot this information. Let's go ahead and start by plotting the vertical asymptote. The vertical asymptote is located at X is equal to -4. Next, we have a horizontal asymptote located at Y is equal to 0. This is going to be directly above or on top of the of the X axis. We also have a white intercept located at -1 on the Y axis. Now, we just determined that on the right hand side of the vertical asymptote, the behavior of the graph is going to approach negative infinity. So, if we approach negative infinity on the right hand side of the vertical asymptote, we can draw the graph with respect to the asymptotes. This is going to be the right-hand behavior of the graph. And on the left hand side we know the graph is going to increase to positive infinity on the left of the vertical asymptote. And with respect to the overall asymptotes this is going to be the behavior of the left hand side of the function, and this is going to be the final shape of the given rational function. So, I hope this video helps you in understanding how to approach this problem, and I will go ahead and see you all in the next video.

Graphing Simple Rational Functions

Graph the rational functions in Exercises 63–68. Include the graphs and equations of the asymptotes and dominant terms.

y = −3/(x − 3)

Key Concepts

Rational Functions

Asymptotes

Dominant Terms

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