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EASIER THAN YOU THINK...

Rational Functions

This is a long video. Think of it as a small lecture.

A rational function is actually a combination of two functions. It is the result of dividing one polynomial by another polynomial.

The existence of a polynomial as a denominator in a fraction introduces two problems/opportunities:
► First, there may now be values of x that would cause us to have a zero denominator and, therefore, an undefined function. We need to understand the implications of this … these zeros identify vertical asymptotes.
► Second, because there is a division taking place, the function can behave in predicatable ways for very large values of x. These give us horizontal (and, sometimes, oblique) asymptotes.

Here, I explain the four key concepts (or principles) that you need to know before you can analysing and understanding rational functions. I then explain (with examples) how to find all the asymptotes necessary and give some guidelines (with examples) that will help you graph a rational function.

This is why it is such a long video. So, settle back and listen. I hope it all makes sense to you.

Thank you to Avis who asked me to explain rational functions to her.

Four Steps to Understanding Rational Functions

This is a long video. Think of it as a small lecture.

A rational function is actually a combination of two functions. It is the result of dividing one polynomial by another polynomial.

The existence of a polynomial as a denominator in a fraction introduces two problems/opportunities:
► First, there may now be values of x that would cause us to have a zero denominator and, therefore, an undefined function. We need to understand the implications of this ... these zeros identify vertical asymptotes.
► Second, because there is a division taking place, the function can behave in predicatable ways for very large values of x. These give us horizontal (and, sometimes, oblique) asymptotes.

Here, I explain the four key concepts (or principles) that you need to know before you can analysing and understanding rational functions. I then explain (with examples) how to find all the asymptotes necessary and give some guidelines (with examples) that will help you graph a rational function.

This is why it is such a long video. So, settle back and listen. I hope it all makes sense to you.

Thank you to Avis who asked me to explain rational functions to her.

How to Graph Type 1 Rational Functions

Now we return to rational functions and learn how we can use all the graphing skills and principles that we have uncovered during the last nine videos.

I restrict myself to discussing Type 1 rational functions here ... where the numerator is a polynomial of lesser degree than the denominator. For such functions, the horizontal asymptote is always y=0, as we have discussed previously.

I demonstrate how to graph ten different equations (assuming that the numerator and denominator have been factorised already), and show how the graphing principles apply in each.

I hope you find it interesting and informative!

QUALIFIER: In this series of videos the curve may not exist in all the locations that I describe. If the numerator is sufficiently small, it will ... but, if you think of these graphs as contours of a three dimensional surface, you will see that sometimes the hill-hummock-mountain does not have sufficient altitude to meet the contour line. Because I am demonstrating principles of locating the graphs, I have not taken the time to substitute values to determine whether the contour lines actually exist. This is a potentially time-consuming and difficult procedure (especially with difficult relations) and would detract from what I am trying to achieve and make the videos very much longer!
If you are checking the graphs at www.wolframalpha.com and do not see the same result, try multiplying the numerator by a small coefficient like 0.05 and see if that makes a difference! As always, I encourage you to experiment.

How to Graph Type 2 Rational Functions

This time I discuss Type 2 rational functions at some length.

In this case, the polynomial numerator and the polynomial denominator are of the same degree (highest power of x). The result of this is that, although there is (still) just one horizontal asymptote, it is no longer along the x-axis! It also means that, if there is a 'change of sign' causing an 's-shaped' bend in the graph, it will cross the horizontal asymptote at an easily identifiable point. In both cases, I show the importance of performing a division first.

This video contains a demonstration of how to graph six different equations (assuming that the numerator and denominator have been factorised already), and shows how the graphing principles that we have learned apply in each.

I hope you find it interesting and informative!

QUALIFIER: In this series of videos the curve may not exist in all the locations that I describe. If the numerator is sufficiently small, it will ... but, if you think of these graphs as contours of a three dimensional surface, you will see that sometimes the hill-hummock-mountain does not have sufficient altitude to meet the contour line. Because I am demonstrating principles of locating the graphs, I have not taken the time to substitute values to determine whether the contour lines actually exist. This is a potentially time-consuming and difficult procedure (especially with difficult relations) and would detract from what I am trying to achieve and make the videos very much longer!
If you are checking the graphs at www.wolframalpha.com and do not see the same result, try multiplying the numerator by a small coefficient like 0.05 and see if that makes a difference! As always, I encourage you to experiment.

How to Graph Type 3 Rational Functions

Type 3 rational functions have either an oblique or curvilinear asymptote. It is this feature that we will particularly explore in this video. We will see how all the other features of these graphs (vertical asymptotes, points of intersection, changes of sign) are affected by the presence of an asymptote that is not horizontal!

The reason we get an oblique or curvilinear asymptote is that, in Type 3 rational functions, the numerator is a polynomial of higher degree than the polynomial in the denominator. This means that, when we divide the numerator by the denominator (using polynomial long division), we get a polynomial result that has a degree of at least one!

A result with degree one is a linear function ... a polynomial of degree two is a quadratic ... a polynomial of degree three is a cubic function, etc. Depending on the difference in degree between the numerator and denominator, a Type 3 rational function could produce any of these as an asymptote.

This video contains a demonstration of how to graph seven different equations (assuming that the numerator and denominator have been factorised already), and shows how the graphing principles that we have learned apply in each.

If you wish to watch a selection of the graphings only, then the times and equations are listed here for you ...

01:09 ~ y = (x^2 - x + 1)/(x - 1)
06:15 ~ y = (x^3 - 3x - 4)/(x + 3) with error in graph (sorry)
11:16 ~ y = (3x^3 - 6x^2 + x - 2)/(x^2 + 3x + 2)
18:08 ~ y = (x^3 - 2x^2 + 1)/(x - 2)
29:26 ~ y = (x^3 - 27)/(x^2 - 9)
38:52 ~ (y+2x-3) = -[(x-2)(x+2)^2]/[(x-3)^2(x-6)(x+1)^2(x+4)]
51:32 ~ (y-x^2+4) = [(x-3)(x-5)]/[(x+1)(x+5)^2(x-6)]

I hope you find it interesting and informative!

QUALIFIER: In this series of videos the curve may not exist in all the locations that I describe. If the numerator is sufficiently small, it will ... but, if you think of these graphs as contours of a three dimensional surface, you will see that sometimes the hill-hummock-mountain does not have sufficient altitude to meet the contour line. Because I am demonstrating principles of locating the graphs, I have not taken the time to substitute values to determine whether the contour lines actually exist. This is a potentially time-consuming and difficult procedure (especially with difficult relations) and would detract from what I am trying to achieve and make the videos very much longer!
If you are checking the graphs at www.wolframalpha.com and do not see the same result, try multiplying the numerator by a small coefficient like 0.05 and see if that makes a difference! As always, I encourage you to experiment.

Negotiating Change of Sign and Multiple Asymptotes

In analysing Type 2 and Type 3 rational functions, one normally divides the numerator by the denominator in order to find the horizontal-oblique-curvilinear asymptote. The rational function that remains may (or may not) have a constant as numerator. In high school questions, it usually will.

If it is not a constant, however, it will be a polynomial in x and will therefore vary in value as x varies. This means that the constant we expected to have keeps changing its size and may even have a value of zero or change from positive to negative! All this has implications for how we interpret a function/relation near those critical x-values.

In this video, in order to demonstrate the major effect clearly, I use just one example with multiple horizontal asymptotes. Of course, a rational function will only have one such asymptote and will be much more easy to graph. This principle has been described in previous videos in this series ... but I thought you might benefit from a summary video as well.

Best wishes from Graeme.

QUALIFIER: In this series of videos the curve may not exist in all the locations that I describe. If the numerator is sufficiently small, it will ... but, if you think of these graphs as contours of a three dimensional surface, you will see that sometimes the hill-hummock-mountain does not have sufficient altitude to meet the contour line. Because I am demonstrating principles of locating the graphs, I have not taken the time to substitute values to determine whether the contour lines actually exist. This is a potentially time-consuming and difficult procedure (especially with difficult relations) and would detract from what I am trying to achieve and make the videos very much longer!
If you are checking the graphs at www.wolframalpha.com and do not see the same result, try multiplying the numerator by a small coefficient like 0.05 and see if that makes a difference! As always, I encourage you to experiment.

What Happens if the Numerator Varies With y?

Mathematicians love to ask, "What if?"

In this case, I ask the question, "What if the numerator on the right is also a function of y?" In other words? What if the value of the numerator (and whether it is positive, zero, or negative) is also governed by the y value? What effect will that have on our graphs?

First, we must appreciate that these would no longer be rational functions ... indeed, they will not be functions at all (simply relations)!

I show how to interpret such numerators and show how our graphs are affected by them.

QUALIFIER: In this series of videos the curve may not exist in all the locations that I describe. If the numerator is sufficiently small, it will ... but, if you think of these graphs as contours of a three dimensional surface, you will see that sometimes the hill-hummock-mountain does not have sufficient altitude to meet the contour line. Because I am demonstrating principles of locating the graphs, I have not taken the time to substitute values to determine whether the contour lines actually exist. This is a potentially time-consuming and difficult procedure (especially with difficult relations) and would detract from what I am trying to achieve and make the videos very much longer!
If you are checking the graphs at www.wolframalpha.com and do not see the same result, try multiplying the numerator by a small coefficient like 0.05 and see if that makes a difference! As always, I encourage you to experiment.

Best teacher ever. You make math effortless.
Christopher C (on a CCM YouTube video demonstrating the Product Rule)

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