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Amy has a master's degree in secondary education and has taught math at a public charter high school.
As you progress in your math, you will come across rational functions, functions made up of one polynomial divided by another polynomial, more and more often. This lesson will help you to analyze these rational functions. You will find these skills essential as you continue in your math learning.
So, for right now, imagine that you are explaining how to graph a rational function to a friend. The rational function that you have in front of you is this one:
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You tell your friend that he does not have to worry. It may look hard, but with what you are going to tell him, he will be able to graph this rational function without any problems. You tell him that you are going to tell him how to break apart this rational function into parts that give you clues about the graph.
You begin by telling him about vertical asymptotes, the vertical lines that the graph approaches. You tell him that the graph approaches these lines but never touches them. The way to find out where these vertical lines are is to set the denominator to 0 and solve the denominator. See, if the denominator is 0, then you will have a division by 0 problem, something that is impossible in math. Because it is impossible in math, it is also impossible to graph.
So, setting your denominator to 0 and then solving for x, you find that x = 1 when your denominator is equal to 0. Solving x - 1 = 0 for x gives x = 1. So, x = 1 is your vertical asymptote. This particular rational function just has one vertical asymptote. But it is possible to have more than one vertical asymptote. There is no limit to the number of vertical asymptotes you can have. Since x = 1 is a vertical asymptote, you tell your friend to draw a vertical dashed line to represent this on the graph.
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Since there is only one vertical asymptote, you tell your friend you can now move on to the next part.
The next part involves horizontal asymptotes, the horizontal lines that the graph approaches. You tell your friend that this process is still simple, but it is a bit different. To find horizontal asymptotes, you need to look at the degree or the highest exponent of the two polynomials. If the degree of the denominator polynomial is greater than the degree of the numerator polynomial, then the horizontal asymptote will be at y = 0. If the degrees are the same, then the horizontal asymptote is found by dividing the leading coefficients.
There is one other scenario outside the scope of this lesson that is not covered here, and that is when the numerator's degree is higher than the denominator's degree in the rational function. This results in what is called a slant asymptote. For this lesson, you won't have to worry about this, and it will be covered in another lesson.
In the case of our current example, the degrees are the same, so you would divide the leading coefficients. In the numerator polynomial, the leading coefficient is 4. In the denominator polynomial, the leading coefficient is 1. Dividing these two, you get y = 4/1 = 4 as your horizontal asymptote. Just like with the vertical asymptote, you tell your friend to draw a dashed line to mark this horizontal asymptote.
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Now that you've covered both the vertical and horizontal asymptotes, you tell your friend you are now ready to graph the function.
To finish up, you plot just a few points on either side of all the vertical asymptotes. To find these points, you plug in different values for x into the rational function and then evaluate it. Since the vertical asymptote is at x = 1, you choose x = 0 and x = -5 to find how the graph behaves to the left of this asymptote. To the right of the asymptote, you choose x = 2 and x = 5.
Plugging in x = 0, you get 0 as an answer. The first point is at (0, 0). Plugging in x = -5 gives you 4 * -5 / (-5 - 1) = -20/-6 = 10/3. The second point is at (-5, 10/3). Next, at x = 2, the function gives 4 * 2 / (2 - 1) = 8/1 = 8. The third point is (2, 8). At x = 5, the function gives 4 * 5 / (5 - 1) = 20/4 = 5. The fourth point is (5, 5). Plotting these on the graph and connecting the dots with a curve that approaches the asymptotes gives us this graph:
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You tell your friend the reason the graph looks like this is because you can't cross the vertical asymptote. That's why the graph is split the way it is. Then you tell your friend that's it! You have graphed the rational function and can now breathe and relax. Maybe even go and get an ice cream.
Let's review what we've learned. In this lesson, we learned to graph rational functions, functions made up of one polynomial divided by another polynomial. There are three steps.
The first step is to find the vertical asymptotes, the vertical lines that the function approaches. To find these, you set the denominator equal to 0 and then solve the denominator. These x values then give you the vertical asymptotes.
The second step is to find the horizontal asymptotes, the horizontal lines that the function approaches. To find these lines, you look at the degrees of the polynomials. If the degree of the denominator polynomial is greater than the degree of the numerator polynomial, then the horizontal asymptote is y = 0. If the degrees are the same, then the horizontal asymptote is found by dividing the leading coefficients.
The third step is to plot points on either side of all vertical asymptotes to find out how the graph behaves in each of these intervals. Then you draw a curved line that connects all your points and that approaches all the asymptotes. Remember, you can't cross a vertical asymptote.
Following this video lesson, you should be able to:
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