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 = 1/(x − 1)

Accepted Answer

Hello, in this video, we are going to be sketching the graph of the given rational function, and we are going to graph the equations of the asymptotes. So the equation given to us is Y is equal to 2 divided by X minus 2. In order to start this problem, let's go ahead and calculate the vertical asymptote. Now, in order to find the vertical asymptote, we want to find the value of X that causes the denominator to be 0. Division by 0 gives us an undefined value, therefore, giving us a vertical asymptote. We will take our denominator, X minus 2, set it equal to 0, and solve for X. And when we do this, we get a vertical asymptote of X is equal to 2. Now what about any slant or horizontal asymptotes? Well, since the degree of the numerator is less than the degree of the denominator, we don't have a slant asymptote. But instead, what we can do is we can try to find the horizontal asymptote. In order to find the horizontal asymptote, we are going to take the limit as X approaches infinity of the given rational function. That is going to be the limit as X approaches infinity of 2 divided by X minus 2. If we direct substitute the limit, we get 2 divided by infinity minus 2. Infinity minus 2 will give us the value of positive infinity, leaving us with 2 divided by infinity. And any number divided by infinity is going to be 0. So that means that we have a horizontal asymptote located as the graph Y is equal to 0. Now, let's go ahead and solve for any intercepts. We will start by solving for the X intercept. In order to find the X intercept, we want to find the values of X that cause the numerator to be zero. But since there is no value of X that causes the numerator to be zero, there is not going to be any X intercepts. Now, what about the Y intercepts? In order to determine the value of the Y intercept, we want to plug in the value of 0 into our graph. That is going to give us Y is equal to 2 divided by 0 minus 2. This will simplify to be 2 divided by -2, which will further simplify to be -1. So we have a Y intercepted located at 0.1. Now we need to determine the behaviors of this graph around the vertical asymptote. In order to do this, we can use the numerical method and check the left hand and right hand limits. So if we take the limit as X approaches 2 from the left of 2 divided by X minus 2, we are going to plug in some values for X that are slightly to the left of the value of 2, and determine a pattern in the outputs of Y. Some values we can pick to the left of two are going to be 1.9, 1.99, and 1.999. When we plug in 1.9, we get the value -20. When we plug in the value of 1.99, we get -200, and when we plug in 1.999, we get negative 2000. As we are getting closer to the vertical asymptote from the left hand side, the graph is decreasing infinitely, and that means the graph will be approaching negative infinity to the left of the vertical asymptote. Now we are going to repeat this process, but when we are approaching the vertical asymptote from the right, So what we are going to do is we are going to plug in values of X that are slightly to the right of the vertical asymptote and look for any behaviors. Some values we can pick is 2.1, 2.01, and 2.001. When we plug in 2.1, we get the value of 20. When we plug in 2.01, we get the value of 200, and when we plug in 2.001, we get the value of 2000. Notice that as we are getting closer to the vertical asymptote from the right hand side, the value of the graph is increasing infinitely, and that means the value of the graph is going to be positive infinity to the right of the vertical asymptope. Now that we've determined these behaviors, the asymptotes, and the intercepts, let's go ahead and plot this information. So, first, we have a vertical asymptote located at X is equal to 2. Next, we have a horizontal asymptote located at Y is equal to 0. That is going to be directly on the X-axis. Next, we have a Y intercept at 0.1, and we have no X intercepts. Now, we just determined that as the graph is approaching the X intercept or the vertical asymptote from the left, the graph is going to decrease infinitely. So we know the behavior of the graph is going to negative infinity when it is on the left of the vertical asymptote. And with respect to the Horizontal asymptote, this is going to be the behavior of the graph to the left of the vertical asymptote. Now, on the right hand side of the vertical asymptote, the graph is going to positive infinity. So with respect to the asymptotes, this is going to be the behavior of the graph to the right of the vertical asymptote. And this is going to be the shape of the given rational function. So I hope this video helps you in understanding how to graph this type of function, 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 = 1/(x − 1)

Key Concepts

Rational Functions

Asymptotes

Dominant Terms

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