How to Graph Polynomials and Construct Their Equations From Graphs
The nice thing about factorised polynomials is that they are quite easy to graph.
First, some terminology needs to be clarified. The roots of such an expression are known as the zeros of the equation (when the polynomial is equal to zero) and these are known as the x-intercepts of the corresponding graph. So, (x-1)²(x+3)(x+5)³ has roots at x = 1, -3 and -5; (x-1)²(x+3)(x+5)³ = 0 has zeros at x = 1, -3 and -5; and, when we graph y = (x-1)²(x+3)(x+5)³, the x-intercepts will be found at x = 1, -3 and -5. I have been guilty of not using this terminology strictly as, in high schools, the terms are often used interchangeably. Perhaps I should be more precise, but there are enough battles to fight in the schooling system.
The factors provide information about where the zeros/roots/x-intercepts are, as well as details about their nature (whether they are single, double, or triple roots, for example).
If a polynomial function has already been factorised, identifying its roots and producing a quick sketch of the graph is quite easy using the skills you will discover here. In the next videos/posts you will see how to handle repeated roots and negative coefficients. After that, I will demonstrate how this collection of curve-sketching skills can be used to create quick preliminary sketches of the graphs required for the Higher School Certificate Examinations in New South Wales, Australia.
Of course, if the polynomial function has not been factorised, you need to learn how to achieve this goal! The first lessons that I will provide will be based on the Factor Theorem, the Remainder Theorem, and Polynomial Division. I will add material about these matters as I am able.
Some graphs of polynomials rise rapidly as you move to the right along the x-axis, and others descend rapidly.
In this video we discover that this property is governed by the leading coefficient (the coefficient of the largest power of x ... i.e. the term with the highest degree). If the leading coefficient is positive, the graph rises to the right; if the leading coefficient is negative, the graph descends to the right. The good news is that ,whether a polynomial is factorised or not, it is quite easy to determine the sign of this coefficient.
This key piece of information, plus a knowledge of the roots, allows you to determine the general shape of the polynomial.
If the leading coefficient of a polynomial is non-monic (that is, not equal to one), then the effect of this number is to increase or decrease the steepness of the curve.
Whether a polynomial is completely factorised or not, it is very important to determine the value of the leading coefficient. Large numbers indicate that the graph is very steep and smaller numbers indicate that it is more flat. This key piece of information, plus a knowledge of the roots, allows you to determine the general shape of the polynomial.
If factors are repeated in a polynomial, that is, if they have degree greater than one, then the corresponding roots are repeated.
We discover in this video that factors with an even degree cause the polynomial to behave 'like a parabola' (a U-shape) near the corresponding x-intercept, and factors with odd degree behave rather 'like a cubic' (an S-shape) near their corresponding x-intercept.
So far, we have learned how to sketch polynomial functions simply by understanding a few principles concerning their roots and leading coefficient. Now it is time for me to demonstrate this set of skills.
As students complete their thirteen years of schooling in NSW, Australia, they sit for the Higher School Certificate examinations. With few exceptions, each year one of the questions in their mathematics paper requires that they sketch a polynomial equation. These questions are primarily designed so that they can demonstrate how to use calculus to determine the curve's stationary points and points of inflection (inflexion in the UK).
It will help you enormously if you can sketch the general shape of the curve within seconds ... before addressing all the parts of the question that seek details about the curve. In this way you will know where to expect stationary points and points of inflection. Therefore, you will be in an excellent position to identify any careless errors that you may make with the detailed work (calculus). If your final graph conforms to the general shape of your original draft, this will also provide you with a boost to your confidence. That can help reduce your stress levels during the examination.
Because this is such a useful group of skills, I have chosen to demonstrate them by taking seven questions from recent HSC examination papers and graphing each one. The HSC Mathematics Examination questions that I discuss in this video are:
2003 Q05a f(x) = x^4 -4x³ (also 2007 Q06b)
2004 Q04b f(x) = x³ - 3x²
2005 Q04b f(x) = (x + 3)(x² - 9)
2006 Q05a f(x) = 2x²(3 - x)
2008 Q08a f(x) = x^4 - 8x²
2010 Q06a f(x) = (x + 2)(x² + 4)
2012 Q14a f(x) = 3x^4 + 4x³ -12x²
Note that, because the 2010 question contained a quadratic factor that did not have real roots, the graphing skills described in these videos were not particularly relevant or helpful. In other words, although these are very powerful tools/skills to have, they are not always successful in helping to graph the function (e.g. if the factors are not linear).
When I was at school I wanted to know how to construct my own formulae. Eventually, I learned a few clever techniques, and this is one of them.
If you know the x-intercepts and y-intercept of a polynomial (from a graph), it is quite 'easy' to deduce the basic polynomial function that generates that graph. The method is simply a reversal of the process that we have discussed in the last few videos. Because of this, learning to create formulae in this way will help you to consolidate your understanding of how to sketch polynomials. You have to use the same key insights in order to find their formulae!
Strictly speaking, polynomials do not appear in this video at all! I have included the video here because it is a 'fun' way of exploring the concept of zeros of an equation (most of the videos about polynomials, so far, have been based on the concept of zeros).
On this occasion I show that it is possible to use the concept of zeros (when the product of a number of factors equals zero) to produce graphs composed of many component parts. In particular, I show how to create just one equation that produces the words I LOVE YOU on graph paper. This might be a very 'nerdy' way to express your love for someone!
Click on the image at right so you can see and print the equation. You will find appropriate graph paper here.
Whne you have finished, you might now ask another question: what kind of graph is created if the product of factors is not equal to zero? I will be answering this question when I eventually produce a series of videos discussing asymptotes.
Although polynomials do not appear in this video at all, I have included the video and downloadable material here because they illustrate how zeros can be used to construct equations. So, this is meant to be a 'fun' and also a rather serious video at the same time. If you are looking for a really unusual and 'nerdy' way of proposing to your girlfriend, and you both share a passion for mathematics and graphing, then this may be what you are looking for.
This idea was inspired a little bit by an unusual proposal that I saw on YouTube where a young man proposed in binary code.
On a more serious note, you will notice that this video is quite long. This is because I needed to review the three graphing skills discussed in previous videos:
how to modify the equation for a circle to create long, thin ellipses,
how to move any graph to new locations on the coordinate plane, and
how to create one large equation that is composed of many smaller ones.
In the video you will discover how to create just one equation that produces the words WILL YOU MARRY ME? on graph paper. Click on the image at right so you can see and print the equation. You will find appropriate graph paper here.
Enjoy and learn!
Download the full instructions (including equation and graph paper) in PDF format here or, if you want to give the material to someone without their being aware of what they are graphing, download the same instructions without the first explanatory page here.
Note: If you are sharing the PDF file, and not a printout, then make sure you change the filename first because it contains the text "WILL YOU MARRY ME?"
Have you ever wanted to create formulae for your own graphs?
This is your chance to develop some very powerful mathematical skills based on a simple principle. The principle is that any number multiplied by zero equals zero! We learn and understand this at quite a young age. In fact, 'zero' would rank as most people's favourite multiplication table!
Surprisingly, this very simple concept allows us to construct the most amazingly complex equations (and their graphs) without much effort. Take the time to watch this video and then download the PDF workbook and see if you can complete the twelve questions contained therein! Believe it or not, if you understand this concept well, then you will be able to understand the behaviour of asymptotes and hyperbolae quite well.
Enjoy and learn!
I have been watching your “Integration technique” videos with high expectation and excitement. They never fail to give me a lot of joy and much pleasure. You must have been one hell of teacher at your time! … just continue making highly enjoyable and intellectually stimulating videos. … Unfortunately, what you learn at the university, you hardly ever see again in your office job. That is why I try to repeat some thing I learned: to enjoy it a bit more, to marvel upon beauty, and to keep my brain sharp. There is where your videos come in perfectly: interesting, amusing and thought provoking problems neatly presented. … You do your best to share your passion, knowledge and experience. The least I can do is to say an honest “Thank you!” As I said, your videos give me much pleasure. You may freely let other people know that. … Rest assured that I shall be watching your videos in the future. With joy and excitement. And I shall comment on them from time to time. You just continue making them.
Zoran V (in private correspondence via YouTube, quoted with permission)