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College Algebra: Info from a Parabola's Equation

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  • Type: Video Tutorial
  • Length: 4:57
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 53 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)
Trigonometry: Conic Sections (12 lessons, $26.73)
Trigonometry: Parabolas (4 lessons, $5.94)

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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So, I just want to show you in a very, very special case how you can actually find the focus. That's that point where if you put a light source there and if the parabola was actually reflective all the light would shine straight out, and also the directorix, which is that line which together with the focus actually produces the parabola. So, suppose you have a very special parabola of the form y = cx². Where c is some number. Like maybe its 5x² or x² or -2x². Well, of course, this is a parabola that has the basic shape--looks like this. Let's pretend c is positive so I'll make it happy face. It has a basic shape that looks like this.
Of course the tightness of this or the looseness depends upon the value of this. If this value, for example, was very, very large--remember, a while ago we saw this would tighten up the parabola, make it very extreme. If that value were very small, then it would sort of make it more relaxed and so you'd have that kind of thing. Well, if you tighten then the focus will actually move up a little bit. If you relax the focus would move down. So, where in fact is the focus? Well, it turns out you can actually now find these points, namely the focus and the directorix, which is going to be the same distance away but on the other side, for this special thing. I'm going to tell you what the answer is right now.
It turns out that if you have y = cx² then the focus is going to be located (0,^1/[4c]). So, that's the point (0,^1/[4c]). This value, therefore, is going to be the line, so the directorix is going to be at the line y equals minus that. So, for this very special type of parabola it's easy to report the news here. For example, let's suppose we have y = ^1/[8]x². Well, then that would bow it out a little bit more. This point should fall. So, let's see what the focus is. Well, the focus would be just what? Well, I take ^1/[8] and put it in here for c and so I'd see one over four times an eighth. Four times an eighth is just a half. So, I have one over a half. One over a half is just two, if you invert and multiply. So, the focus would be located at (0,2).
What's this value? Well, this is going to be -2. So, this is going to be -2. The same distance up, the same distance down. That's how you can find the directorix if you want to. The exact same rules hold if the parabola is actually this way. Let me just show you that really fast. So, if you have something like x = cy², that's a similar creature, but now it opens this way if c is positive or this way if c is negative. Then, it turns out you could find the focus and the directorix in a similar way. The focus, it would be located at what point? Well, it would be located at (^1/[4c],0) and the directorix now it's going to be an x value. It's going to be x = -^1/[4c].
So, for example, if I had x = -4y² look what that would mean. A minus sign means it would open this way. If it opens this way, this focus should be negative and the directorix should be positive. Let's see if that really happens. Well, the focus would be at what? Well, I put in a negative four for c and so I'd see one over -16 or (-^1/[16],0). Notice that is negative and the four means it's very, very tight. It's very, very tight, that means that this focus is going to slip way far in there. What about the directorix? The directorix would be at x equals the opposite of this. Which would be ^1/[16]. So, now the directorix would be on this side. So, it'd be the exact flip of this kind of thing.
So, if it's a parabola that opens up this way, the focus will be in here and the directorix would be out here. Neat, not a problem, you can always find the focus ad directorix of something of this form very easily, by just looking at the coefficient and producing ^1/[4c] and then the directorix would be at -^1/[4c]. That's all there is to it. Pretty simple.
Conic Sections
Parabolas
Determining Information about a Parabola from its Equation Page [1 of 1]

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