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College Algebra: Graphing Polynomial Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 17:06
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 182 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra Review (30 lessons, $59.40)

In this lesson, you will learn how to graph polynomials on the Cartesian coordinate plane (without using a graphing calculator). Professor Burger will show you how to identify the degree of a polynomial (first-degree, second-degree, third-degree, etc) and how to predict what a graphed polynomial will look.

This lesson is perfect for review for a CLEP test, mid-term, final, summer school, or personal growth!

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

2174 lessons

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

Thinkwell lessons feature a star-studded cast of outstanding university professors: Edward Burger (Pre-Algebra through...


Recent Reviews

Graphing without a calculator. Wow!
~ nachan

What an amazing professor Burger! I never realized that you can graph polynomials with out using a graphing TI calculator! He explains how to sort problems in factor and non factor form and teaches you how to foil binomials. Nice length 17 minutes video to help thoroughly explain graphing polynomial functions.

Graphing without a calculator. Wow!
~ nachan

What an amazing professor Burger! I never realized that you can graph polynomials with out using a graphing TI calculator! He explains how to sort problems in factor and non factor form and teaches you how to foil binomials. Nice length 17 minutes video to help thoroughly explain graphing polynomial functions.

Alright so you know graphing polynomial functions serious business but there are methods, there are steps we can take to actually have us kind of go through and systematically figure out what we think in fact the function might look like in order to sketch a reasonable sense of what the graph might be associated for a polynomial. And I want to illustrate this by taking a look at a couple of examples together and just kind of talking you through some of the steps that you might want to consider taking when you are actually looking at polynomial functions and you want to sketch a rough graph or whatever it might be. So here is a function right here f(x) = x(x+4) (x-3). Now I must say that I really like this particular function. Why, because they gave us the polynomial in factored form. That’s going to make our lives a lot easier because I can immediately read off all sorts of information that I will talk about in a second. But suppose this wasn’t in factor form or suppose that we for some bizarre reason which makes no sense want to go and foil it all out. What degree of polynomial would this be? Well let’s take a look at the highest power of x. So even I am not going to foil the whole thing out I want to just find the world’s highest power of x in this particular function. Well when I foil out this binomial I am going to have an (x)2 here and then I am going to multiply through by this outer x here and so this is actually going to be a cubic and you will notice that in fact since it’s a cubic and there is no other higher terms this is a third degree polynomial and so it genuinely is a cubic. Now how do cubics look? Well cubics have a basic flavor and basically they have two kind of turning points here and so cubics will kind of have the following type of flavor. Now it might be something like this. That could be a cubic, might be a lot more subtle. It might be something even as subtle as this, two kinds of turning points, two points for turning or it might be something as subtle as this where it doesn’t even seem to turn it off but in each of these cases you notice something interesting. You notice that we start way, way down and we kind of work our way up. This is always true when we look at the coefficient on the (x)3 term and see a number that’s positive. So if the leading coefficient which in this case is coefficient is the (x)3 term is positive then what it means is that when we move off to the left it’s going to get very, very, very negative. We move off to the right horizon it’s going to get very, very, very positive. Notice that all three of these graphs have that feature. This is something that looks like a positive number times (x)2 where this thing is a positive number and in this was a cube rather and then these other lesser terms they can be positive or negative. The leading term tells me the kind of what’s happening as we go off the horizon. If in fact this number were negative then in fact these pictures would look in some sense like the mirror images of themselves so they would look like this. They would look like this or this or this and you can see here that as the x moves off to the left now the ys get bigger and as the x moves off to the right the ys get smaller. This would look like something where the leading coefficient is actually negative and then you have the other terms so just by looking at that I can tell what rough shape this is going to be so I see the coefficient here is 1x1x1 the leading term is (x)3 there is an invisible 1 in front so I know it’s not going to look like this in fact it’s going to look like one of these. Okay now where do I go from here? There are a lot of different versions of these and also there is a lot of placement that this could be actually sitting in the plain so how do I figure out which one is appropriate. Well what I will do now is actually get some information about this thing. One thing I can do which is really easy is to find the y intercept. Now the y intercept is always an easy point to find. That’s where the function crosses the Y-axis of course you know we cross the Y-axis when x=0 the equation of the Y-axis after all is x=0 so this will let x=0 that’s just f(0) so that x=0 and see what you get. You get 0x(0+4)x(0-3). Well 0 time anything is 0 its 0, 0, 0 so 0,0 is the Y intercept that’s awesome so that’s a point that we now know is on this particular graph of this particular polynomial. Another great thing to do if we can is to see if we can find the, I am sorry so that was the Y intercept. Now we can also try to find the X-intercepts. That’s in fact where y=0 so I want to set this equal to 0. So if I set this equal to 0, since it’s so conveniently in factored form I could literally read off the answers. This could be 0 which means x=0 which is what we just found here or the possibility. So x=0 is one possibility, one we already have. Another one is that x+4=0 which means x=(-4). Another possibility is x-3=0 which means x=3. So these are all the X-intercepts. So these are the X-intercepts, this is the Y-intercept we found out when of course they shared that point the origin in comment and so that’s awesome. So that actually allows us to kind of in some sense lock down, pin down if you will on a few points from this graph. You might want to find some other points which is totally fine. You might want to ask if everything is symmetric around the Y-axis and so forth and those little things of that sort we can do, but now let’s start plugging in these points and seeing if we can make a guess as to oh, and I actually look what I found, pair of axes, no waiting. So here we go, beautiful, beautiful. I am loving. Okay so let’s plug these points. So I have 0,0 as the Y-intercept that version is across the Y-axis so that’s 0,0 and I see them across the X-axis and x=(-4) so x=(-4), it’s been right here so went across the X-axis again there and then at x=3 we cross the X-axis again. Well if we are going to cross the X-axis at three points and we know that the leading coefficient is positive which of these versions could it be. Well if here’s the axis the X-axis, could it be this version. No, because that’s only going to cross the X-axis at one point. Could it be this version? No, but this version notice could cross at three points. So this is going to be a function that’s going to climb up then peak somewhere, climb back down then have kind of a low-low point and then climb back up. It’s going to be this picture and we see that just by this information right here. Now if you want you can plug some more points which might be kind of fund, if you can just plot those points by you know picking nice points plugging them in. For example, you might want to pick x=1. Suppose we picked x=1 just for fun. We have got x=1 then what would we see. We would see that y would equal 1 times 1+4 that’s 5. 1-3 is (-2), (-2) and 5 is (-10), (-10) times 1 is (-10) and so what I would see here is that at 1 I am at (-10). You can plug in 2 and see what you get. You can plug in (-1), plug in (-1) you are actually going to get what, I think you are going to get 8, 9, 10, 11, 12, I think 12 I guess and so on you can plot some more points. And now this really actually helps us to see that our initial guess is looking good, is going to make form to this. It’s not that I am going to try to draw it in. I don’t exactly know where that maximum point is for sure but I do know it’s going to have this nice kind of feature where I am going to shoot up and then come up, come down, downtown, come around and then come on up, come on up, like this. This is not so good. I have to get myself through with this. This shouldn’t be wiggling at all. There we go. The way you can get rid of your wiggles by the way if you are doing this for an actual homework is to make the ink flicker and you can flick it out and it wiggles. And notice this really has that look that we had before, like that, point, there is the graph of this particular cubic equation, pretty cool, pretty cool. Alright let’s try one more together. This do require a bit of work by the way. You could see this or not you know just a quick you know looking graph. This is a little bit of work but all worth it. Let’s take a look at this. What kind of polynomial equation, polynomial function is this. Well you will notice that when I foil this out it’s going to be a square term so it’s erratic plus when I multiply through by this particular x I am going to get a cubic so this is a cubic as well. And what’s the leading coefficient? The leading coefficient here is 1, when it’s squared, it’s 1, 1, 1, I get an x3 but there’s a negative sign way out in the front. This is a negatively, it has a negative coefficient which means that its picture is going to look something like one of these and we don’t know which one though. So let’s see if we can actually find some nice points. First of all let’s find the Y-intercept. The Y-intercept is when x=0 so you plug in 0. I see 0+12 0-2 and so what do I see. I see this is just 1. This is a (-2) and negative sign is just 2. So I see that 0,2 is going to be the point that we cross, in fact the Y-intercept is at 2. So that’s the point I am just going to lasso that. What about the X-intercepts? That’s where it crosses the X-axis and happily again we see it’s in factored form so this is going to be really awesome. We see that either this factor is equal to 0 which means that x+1=0 or x=(-1) or this factor equals 0 which means x=2. Now, you will notice that there are only two solutions to what when I set this equal to 0. Technically there is always going to be 3 solutions to any cubic polynomial equation so there must be an invisible third solution that we haven't found yet. Where is that? Well actually if you were to write this out we would actually write out –(x+1)(x+1)(x-2), namely I have this appearing twice. So therefore I would actually have the invisible solution which is a repeat of this solution. So in some sense I have the solution x=(-1) with multiplicity 2. That means something weird is going to happen there. What it means is well, let’s see what it means. Something interesting is going to happen. Again we can now take a look at symmetric. Again you can ask for symmetry questions. Those are always great fun. If it’s symmetric and on Y-axis, if you replace x by (-x) if the thing is the same it’s symmetric around the Y-axis like a mirror image. This is not like that though so it’s 1. Alright, if x=0 y=2, so if x=0 y=2 we get this point right here and then we have these X-intercepts at (-1) which happens twice, which means nothing in terms of drawing the dot and also at 2. Now this is a little weird because we know we have a negative coefficient so we know it’s going to be one of these, but which one of these is it going to be? This is not at all clear and of these three choices. Take a little kind of puzzle. So if you are not sure the best thing to do is to just plot maybe one or two more points just to get a feel of what’s going on. Let’s for example let x=1 see what happens. So if we let x=1 what’s f(1), well f(1) will be negative (1+1=2) so I have 22 times and I then I have putting 1 here, 1-2=(-1). And so what do I see, I see a 4(-) (-) is a positive so this is 4. So I see when at 1, at 1 we are at 4 so we are way up here like this. Does that give us a better clue? Well it certainly tells me that what’s going to happen here is this must be the upward swing and this must be the down swing here, right. So maybe what we should do is figure out you know where is that turning point thing. Well let’s take a look at (-2) so what’s f(-2) that’s going to be (-)x(-2+1) (-2+1) is what it’s (-12 times and -2-2 is actually -4. Well (-1)2 is going to be 1 and negative and negative is a positive so this actually is 4 again that’s kind of amazing. So here we bounce and we are at 4. And it turns out in this example this is going to be the low point. So the two points that we see here with multiplicity 2 actually come from this configuration. So in some sense the graph comes down and just kisses the X-axis. And that turn is while we get two solutions and you can see this for yourself, because what if I just jiggle the X-axis, oops see automatically we see now the two-solutions, but kind of in this limiting case the two-solutions become one-solution. And so there is one-solution here and one solution here this is the upswing that’s the part we are seeing here, this part here is this part. And so all of a sudden I can draw wonderful picture of this and it looks like this. Come down nice and steep, come way down, grab that point and then make a quick turning point it’s a minimum, climb back up, I don’t exactly know where the max is, it might be at 1 but it might not be. And then come on back down. And notice that it has that general form of a cubic that has a negative coefficient and we have a rough sketch of this. So you can produce this rough sketch, absolutely unique. Well you might say well I don’t exactly know where that highest point is, and I don’t exactly know how this you know curvature of all this stuff goes and how would I find that, well it turns out by plotting points we can’t exactly guarantee exactly where all those little nuanced elements are. Now if you have software you can go on a computer and they can actually produce this really-really beautiful pictures but knowing for sure that that software is correct, actually is something is little tricky and you want to know what the answer, the answer is the techniques of Calculus. It’s in Calculus we will finally see why these curves, curve the way they do and where that curvature really is. Those peaks and those valleys can be found exactly using the ideas of Calculus but for now we can celebrate the fact that by solving you know the equation in terms of finding the X-intercepts the Y-intercepts looking for symmetry, plotting a few points and thinking about what’s happening as you go off the horizon whether it is going like this, like this for cubics for example we can actually get a very good rough sense of how this graphs look. Congratulations on thinking about these wonderful graphs. Have fun thinking about them, lots of steps involved but you absolutely can do it, I know it, I will see you soon.

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