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College Algebra: Quadratic Function Graph

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About this Lesson

  • Type: Video Tutorial
  • Length: 5:29
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 59 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Trigonometry: Full Course (152 lessons, $148.50)
College Algebra: Relations and Functions (57 lessons, $74.25)
Trigonometry: Algebra Prerequisites (60 lessons, $69.30)
College Algebra: Quadratic Functions: Basics (4 lessons, $4.95)

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 athttp://www.thinkwell.com/student/product/collegealgebra. 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.

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Thinkwell
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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 www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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So now I want us to think about one of my favorite functions in the world after the straight line functions, and these are the quadratics. So now it's time for some quadratic functions. That's right, folks. It's time to look at quadratic functions. These are the functions that actually graphically represent parabolas. So this is really great, because of course, a parabola is so neat. If you're not going to be a line, you might as well be a parabola, because you have your ups and downs. It's like life, you know? You go down and you go up. Or maybe you go up and then you go down. The great thing about parabolas is first of all, there's incredible symmetry there. Isn't that neat? In fact, if you had a mirror you can really just see--look at that. There it is. Just a mirror, it gives you the whole thing. See? Reflection, real thing, reflection, real thing.
Anyway, the point is there's actually a symmetry here. There's actually an axis of symmetry, an axis where, in fact, if you go along that axis and flip it, you actually get a symmetric thing. So the parabola is a very symmetric curve. What you do on the right hand side of the axis is symmetry as what you're going to do on the left hand side. Also, you'll notice the parabola has a feature that there's always going to be one point where things turn around--a turning point. This is called a vertex. In this case, I go from going down to going up. So this is a minimum--this is a turning point that's a minimum, so this is a vertex that's a minimum, so you can imagine a parabola that looks like this--first you go up and then you come down. That vertex is actually a maximum. So, in fact, there's this one vertex, which you should be able to define, and then we have this axis of symmetry, which is pretty neat, and so we can talk about domains and ranges and all sorts of neat stuff like that, but in fact, you can actually start to graph these things pretty easily.
For example, let me take a look at a particular graph. Let's consider the following function. I don't know if you can read that or not. f(x) = (x + 3)^2 - 4. Now, the graph of it actually looks like this, so in fact, the vertex here is located at (-3,-4). That's the vertex. Now, actually we could graph that using some of the techniques of graphing that we've seen by shifting and so forth, and translating.
Let me show you that real fast, because we all know what the standard parabola looks like. The standard happy parabola looks like this. Now, what do we do? (x + 3) means I'm going to be shifting in the x direction. So that's going to be a right or left shift? Do I go right or do I go left? Well, always remember... People like to say, "Oh, I'm going to go right because it's plus 3, I'm going to add 3." That's a classic mistake. In fact, this is classic mistake number 8 in my top ten list of classic mistakes. That's right, number 8. If you add to y, you'd better go high. If you add to x, you'd better go west. The shifting function mistake. And always remember the slogan, add to y, go high; add to x, go west. So if we're going to add 3, that means we should go west 3 units, so go west 3 units. So go to the left one, two, three units. Okay, so that's the (x + 3)^2 part. What about the -4? Well, y go high, so therefore, what do I do? If I add to y, go high. Minus 4 means I should go down since I'm subtracting by 4 units, and there you see the picture that we have here. So, in fact, this makes a lot of sense. It's just a good, old-fashioned parabola that's been shifted and shifted. So, in fact, hidden in here, we can actually see the coordinates for the vertex. The coordinates for the vertex turn out to be at (-3,-4).
Now, we can ask other questions. For example, we could ask what is the domain of this function? What are all the allowable x values? Well, it turns out for quadratics the domain is everything, because you could take any value at all and always plug it into a quadratic, because a quadratic, after all, is just something times a quantity squared plus some other number times the quantity, plus some third number. So you can always do that with no division with quadratics. So, in fact, the domain is everything. What about the range? Well, let me remind you, the range, those are all the possible y target points you hit, and you'll notice in this particular problem the range, well, those are all the values that are -4 or above, because all those points will get hit by some point, right? You can see these blue things, if you can sort of super-impose them onto the line, will hit everything here. But there's nothing down here. The graph never dips below this line. So, in fact, the range would be from -4 up to infinity for the y values. That would be the range, and the domain, of course, is everything.
Okay, so what I want to take a look at next is to figure out how we can just look at a very nice formulation of the parabola, of this form, and realize where the vertex is. And the vertex, you can see here, is at (-3, -4). So that's a hint as to how to figure out the coordinates for the vertex. We look sort of at this number--looks like we take the negative, and then we take a look at this number, we put them together, and that should actually give us the coordinates of the vertex. Once we have the vertex we know the axis of symmetry and know if it goes up and over like this, or maybe it's a sad face parabola. We'll take a look at those parabolas and how to actually determine the vertex just looking at the equation coming up next. I'll see you there.
Relations and Functions
Quadratic Functions- Basics
Deconstructing the Graph of a Quadratic Functions Page [1 of 1]

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