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About this Lesson
 Type: Video Tutorial
 Length: 6:58
 Media: Video/mp4
 Use: Watch Online & Download
 Access Period: Unrestricted
 Download: MP4 (iPod compatible)
 Size: 74 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: The Vertex (4 lessons, $6.93)
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 UTAustin 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, padic 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
 Thinkwell
 2174 lessons
 Joined:
11/13/2008
Founded in 1997, Thinkwell has succeeded in creating "nextgeneration" 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 technologybased 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 if you have a parabola, if someone actually gives you a parabola, we've already talked about the fact that it sort of has this point where it's either going to be a minimum, in this case, this vertex will either be a minimum if it's a happy face parabola, or if it's sad face parabola, we're going to have a maximum. And the question is, how can you actually find, first of all, that maximum value, in this case, and then where that maximum value actually is taken on, for what x value.
Similarly, if you've got a minimum, what is that minimum value, and where does it happen? So that really is asking nothing more and nothing less than asking, where is that vertex point? So if you want to find that vertex point, that vertex point will always be either the maximum or the minimum value of the parabola, depending upon whether it's a happy face parabola or a sad face parabola. You see, in this case, if this is, let's say, h and this is k, see the vertex is going to happen, in this case, if it's a happy face parabola, it's going to give us a minimum value, and that minimum value is the y value. That would be k. And where that minimum value happens, where that occurs, is going to be h.
Now, conversely, if it's a sad face parabola, suppose it looks like this, well then we have a maximum and that maximum will be at the vertex, and that maximum height, the actual height, the maximum value of this is going to be a k, and the location, where that happens, is actually going to be at h. So if someone says find the maximum or minimum of a particular parabola, all you've got to do is find the vertex and determine whether it's a happy face of a sad face parabola.
So let me try a couple of examples to really illustrate this point. Suppose we're given the following: f(x) = 2x^2 + 3x + 1. What I want to do is first of all determine does this have a maximum or a minimum. Let's think about it. Is this a happy face parabola or a sad face? Well, this is a positive coefficient, so, in fact, it's going to be a happy face parabola. If it's a happy face parabola it will have a minimum. So this has a minimum. You may say, "Oh, it's positive, therefore it's max." No, no, no. Since it's positive it goes down and then up, it's happy faced, and so we have a minimum there. Now, where's the minimum and what's the value of it? Well, that's going to be exactly at the vertex. So let's find the vertex. So to find the vertex we know h is going to be and k is just going to be f evaluated at that point. And so what do we have here? So h is going to be b, that's going to be , and that's going to be 4, so we see 3/4. And now what's k? Well, what I have to do here is plug in 3/4 into here, so I have to do a little calculation. So we have 2 times 3/4^2, which is a 9/16, and then I add 3 times 3/4, which will be a 9/4, and then I have a +1. What does that equal? Well, there's a little cancellation I can do here. This 2 cancels with this and makes that an 8. And so now I have 9/8 and I have to add 9/4. Well, 9/4 is just 18/8, and 18/8 and 9/8 is just going to be 9/8. So this is going to equal 9/8 and then I have a +1 and 1 is 8/8 and 8/8 and 9/8 is 1/8. So the vertex is going to be at (3/4, 1/8). That's the vertex.
So first of all, what is the minimum value of the associated parabola? The minimum value is this y value, the 1/8. That's as low as it's going to go. And where does that happen? What's the location where that takes place? At 3/4. So if you think about it visually, what I see here is the following: I see that 3/4, that's right here, and 1/8, right here, that's where that minimum is. So the lowest value is 1/8, that's the y value, and the location of it, 3/4.
Let's try another example. How about f(x) = x^2 + 4x + 1. Well, first of all, will this have a max or a min? Well, I see a negative coefficient in front of the x^2 so this means this is a sad face parabola, so this will actually have a maximum, and where is that maximum? So this has a max and that maximum is going to be located right at the vertex, so I have to find the vertex. So first I'm going to find h. So that's b, so that's going to be , so that's 2, and so that gives me a net gain of 2. And then what's k, the y coordinate? That'll be f(2) and so what's f(2)? Well, I'm going to plug in 2 here and I'm going to see therefore, 2^2 which is 4, so I see a 4, and then I see a +8, and then a +1. So this gives me +4 and 1 is 5. So the vertex is going to be at (2, 5). So therefore, does that mean the maximum is equal to 2? Is the maximum 2? Is the highest this curve actually gets equal to 2? No. The highest it gets is 5. That's the y value. So the maximum valuethe max value equals 5. And where does that happen? It happens at x = 2. Okay? So if someone says what is the maximum value, the answer is 5. If someone says where does this function attain its maximum value, the answer is x = 2. Graphically, what's going on, well, we have a sad face parabola and its vertex is at one, two, one, two, three, four, five. So there's the vertex. There's the max, and there you can see exactly what's happening. The highest point value is 5 and it's attained at x = 2. So you can actually now find the max or the min of a parabola, depending if it's happy face or sad face, by just finding the vertex and then reading off the result. Try these.
Relations and Functions
Quadratic Functions the Vertex
Finding the Maximum or Minimum of a Quadratic Page [2 of 2]
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