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College Algebra: Sketch Basic Polynomial Functions


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:16
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 121 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Polynomial & Rational Functions (23 lessons, $35.64)
College Algebra: Graphing Polynomials (2 lessons, $2.97)

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

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So, let's try to recap what we just saw. We have a polynomial that has its highest degree being even, then we know the general shape of it is going to be something sort of like this, and then it's going to wiggle something, sort of in between. If it were a negative coefficient in front, its general shape would be this. It would sort of come up here, wiggle a little bit, and go down. If you have an odd highest degree polynomial that you're graphing, then basically what it's going to look like is something like this. Let's see, how would it look to you? It would look like this. So, it would sort of do this kind of thing, wiggle a little bit, potentially, in here and then end up down. Unless it was a negative coefficient in front, in which case it would sort do the opposite and look like this kind of thing.
Now, the very basic, the most simple even degrees--of course, the first one is the parabola and that one just looks like this. So, the is y = x². Now, what would y = x^4 look like? Well, it would be sort of a similar kind of thing, it would come down and go up, but actually what turns out is it sort of starts in a little bit further and then drops. It's still very smooth, but just a slightly different shape. Okay, that's x^4, x^6 would actually even have a more dramatic thing. It would start a little further in, come down, go down a little sharper, but come up--still smooth though, no corners, always very smooth and symmetric. But, you can see what's happening, it sort of converging into something that looks very sharp like this. The higher the degree, the more it looks like that if it's even.
By the way, did you notice that they all contained the same point? If you think about it and check, that point is actually (1,1) and (-1,1). If you put 1 into x², x^4, x^6, you always get 1. If you plug in -1 into x², x^4, x^6, you always get 1 again. So, those two points--of course, they should look even, they should be able to be even looking. Those two points are points in common with all these things. The higher the degree, the more U like, the more U like the thing is going to be.
What about if you have an odd degree? Well, the most classic odd one would be the cubic. I guess actually the most classic odd one would be linear, just the line. That's odd, right? That's just x^1. Then the cubic is more interesting. It has that little wiggle. So, there's x³. Now, what's x^5 look like? Well, it's the same kind of thing as is going on here. It's going to be the same basic shape, but just sort of sharper. So, it's going to start a little taller up and then come down faster, go over and do that. That's going to be x^5. What you can see we're heading toward is something that's sort of looks very, very sharp, comes down and comes down like this. The higher the degree, the tighter it gets around those corners. Always very smooth though, never sharp. Because it's a polynomial, it's always extremely smooth. But, it has that basic shape and you can see these things.
Again, you'll notice that all the points have in common, these two. What are those? Well, at 1, it's 1, because if you take 1^1 or 1³ or 1^5 or 1^7, you always get 1. Notice that if you plug in -1, -1^1, -1³, or -1^5, you always get -1. So, those points are always points in common, with the very, very basic cubic. Just as we have similar points with the even power things. Now, just knowing what these basic, basic shapes look like, you can start to graph all sorts of stuff. Let's do some simple examples to illustrate the point.
So, let's graph f(x) = 2x^4. Well, I know it's going to be a sitting up kind of cup and the 2 in front, if you remember, sort of brings things in, makes things tighter. So, it should look something like that. It should have sort of a parabola looking, parabolaesque kind of thing. We can plot some points really fast here. Plug in 0, we get 0. Plug in 1 and we get 2. You can see it climbs really fast. -1 is going to be also 2. Remember, this is an even functions. It's going to be symmetric against the y-axis. If you were to plug in 2, I'd go way off the screen. Right? Because 2 would actually give me 16. So, that'd be way off the screen. I'd climb really, really fast. A real fast climber. This function is going places. It would look something like that. That would be the graph of it.
Let's try another one. How about ^1/[3]f(x) = ^1/[3]x^6? Well, x^6 still has that general parabolaesque thing, but it's even slightly sharper. Even though it's curvy, it's more dramatic. Straighter here, sharper turn to get to the origin, straighter down here and then a sharp go up like that. Even more sharper than this, but I'm taking a third of it. So, that means--remember what a third does. A coefficient that's less than 1 tends to stretch things out a little bit. So, even though I'm starting off very tight, I'm going to loosen it up a little bit. I should expect that kind of activity.
Again, plotting a couple of points. Not a bad idea. Let's see. If we put in 0, we get 0. That's pretty easy. If I plug in a 1, look what happens. I get just a third. So, there I don't climb too much at all. I just climb right to here. You can really see--and the same thing with -1. It's an even function, again. So, my growth is going to be very gradual. At 2, what am I going to have? Well, I'm going to have 2^6. That's a pretty big number. That's around like 64. So, 64, but then I take a third of it. That's going to be around 20 something. So, by 2 or more, you're starting to grow pretty fast. It looks something like this. Again, has that general parabolic feel, but at 2 I'm at ^64/[3]. You could actually mark that down and say, "Not drawn to scale." Anyway, you get the sense of how these things would bend and curve. Notice, as I multiply by the third out there, it flexes out a little teeny bit, but still growing quite dramatically.
Let's try this one. f(x) = -(x - 1)³. How would we tackle something like this? Well, the first thing I notice--I'm going to do some intermediate steps. In fact, maybe you want to do this kind of thing in pencil or something, or just do it on the side. The first thing I notice is, basically, this is just sort of something cubed. Now, there's a negative sign in front, which mean it's not going to be this kind of cube. Now, watch this. This is just x³. That's x³ right there. It's going to be negative that. So, that's a flip over the x-axis. It's going to look like this. If I draw that in, I'll draw it in in dotted, this is just -x³. What I want is that thing, but I want x - 1, that part cubed. So, what does that mean? That's a shift in the x, which means I'm going to be shifting left or right.
Now, which way do I go? Well, it's tempting to say, "Well, x minus something, I'll shift to the left." Remember, that's a classic mistake. It's a classic mistake number 8. It's the shifting function mistake. Remember, add to y, go high, add to x, go west. So, if you add a number, you should go to the west. So, if I'm subtracting, I should go to the east by one unit. So, I should just take this picture and literally, rigidly just shift it over one. The actual picture would look like this. Exact same picture, but just shifted over one. That's the function f(x). There you get a very accurate picture of it by first plotting this other thing and then just doing a shift.
One last one. How about this? f(x) = (x + 2)^4 - 1. What would I do here? Well, I would do a whole bunch of intermediate steps, maybe. Before I actually report the news. So, the first thing I'll do is just say, "What's x^4? That's sort of the key thing here. Well, that's sort of a parabola thing, but a little bit more exaggerated. A little more dramatic. It's a dramatic parabola. Again, you can plot points, if you don't like thinking about drama--very symmetric. That is just x^4. Not what I want. Okay, then what? Well, then what's (x + 2)^4? That's a shift in the x. Which way? Remember, add to x, go west--so, I'm going to shift it two units this way.
If I shift this whole picture two units this way, it would look something like this. Same picture, just shifted. So, that's (x + 2)^4. Now, I'm ready for the final picture. Then what I do is take everything and subtract 1. So, that's now--I just take everything and subtract 1. Now, I just take every y value and I deduct 1 from it. So, that's going to be our shift down. So, you get down by one unit and then we get the final picture. This picture shifts down by one and it looks like this. I'd actually put in my intercepts, if I can, really, really quickly here. So, when x = 0, this thing is going to be 2^4 - 1. So, some big positive number.
So, it's literally this picture right here, shifted down one unit. Just one unit down. So, this point right here has a height of -1. Again, you an see that the way I graph something like this, it looks pretty complicated, is just to see the essense of it. Graph the essence and I start booting my way up. This is a change in x, two units to the left. So, then this brings me own one. So, it's just an easy picture that we've already seen. That's all there is to it. Try some of these yourself.
Polynomial and Rational Functions
Graphing Polynomials
Sketching the Graphs of Basic Polynomial Functions Page [1 of 2]

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