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College Algebra: Stretching a Graph

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  • Type: Video Tutorial
  • Length: 10:19
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 111 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: Graphs: Shifts and Stretches (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 we see how we can translate functions or shift functions by either adding something to the very end, which is a shift in the y direction, which is a shift in the y direction, which makes this go up or down, or actually replacing all the x's by like x + 3 everywhere, in which case we would shift to the left by 3 units, or x - 3, in which case we would shift everything to the right 3 units. But what if you just take a function that you know and love, like the parabola, you know, f(x) = x^2, and what if you just multiplied it by something. Like what if you looked at 2x^2? What's the effect of multiplication of a function by a number? Well, let's see if we can figure out exactly how that would effect things. So let's actually look at the function f(x) = x^2. We know it's going to be this happy face parabola--not a big surprise. In fact, we know it's always going to have positive y values, since in fact, we know that... Now let's actually try to plot some points just to see. Now, we know exactly what this looks like but I want to be accurate here, so I've got 0, 0. We're at 1, because 1^2 is 1. At 2 we're at 4, so we have 1, 2, 3, 4. So it looks like this, and then we sort of see the parabola. Now, we've got the exact same thing going on in the negative side, so I have -1, 1 and then -2, 4, and if I connect them we should get this beautiful looking parabola. Look how gorgeous. Now drawing live parabolas, by the way, is a lot harder than it may look. You try it. Get a whole bunch of friends together and draw a parabola right in front of them. It's harder than it looks.
Anyway, this is the function f(x) = x^2, Okay great. Now let's consider a different function. Let's consider this function now. Let's consider f(x) = 2x^2. I want to take this parabola and I want to multiply it by 2. Now what's the effect going to be? Well, we know it's not going to be a shift effect, because if we're going to shift it that means we have to add something to this. It's not going to be a slide or a shift this way because that means I replace the x by x + 5 or x - 5 or something. So we're not going to be shifting it. The parabola's going to basically stay in place somehow, but this is going to be some sort of distortion in some sense, and in fact, you can think of this sort of like a stretching or contracting of the curve. And so what's going to happen is the following. Let's just try some examples and plot some points.
If I plug in zero, notice that zero, 0^2 x 2 is still 0. So that point, in fact, doesn't even move. Maybe nothing moves. Let's see. What about at 1? Before I was at 1. Where am I now? Well, now when I plug in a 1 I get 1^2 which is still 1, but I've got to multiply it by 2, so that gives me 2 now. So notice that what's happening is this point at 1 no longer is here, it's now going to live way up here at 2. Similarly, if I plug in -1 and square it I get 1 times 2 is 2, so I'm going to be way up here. So these points now get moved to here. Where would this point get moved to? Well, this point is 2; it used to be at 4. Now where is it going to be? It's going to be a 2 times 2^2, which is 8. So, in fact, this point is going to get moved way, way, way, way up to about here. And similarly, here, this point will get moved way up here.
So what does our curve look like? It sort of looks like the parabola, but I sort of brought it closer together. Those two wings are getting closer together. Do you see how they want to sort of be sharper now? It's going to be a sharper parabola. So what we're going to have here is a much sharper parabola, from the sense that these wings, in some sense, closed up a little bit. It's still the same kind of parabola curve, but now it's in some sense more steep. The steepness has increased, because instead of being out here, it's now here. So when you put in a multiplicative factor that's bigger than 1, what that does is it actually makes things more steep. It makes the steepness greater. So if I have a nice, happy little parabola like that, multiply it by 7, and it's going to be even more steep, so it's going to come down and then shoot up and then come back--be really, really steep. The bigger the number, the sharper the steepness is going to be.
What happens if you do the opposite thing? Let's take a look at what happens if you multiply by a number that's small. How about times x^2? Well, let's make a guess as to what we think is going to happen and see if it actually happens or not. What's a good guess? Well, if I multiply by a number that's bigger than 1 it makes things really steeper. If I multiply it by a number less than 1, in some sense maybe that should make things less steep. Maybe it should actually flare up and make the thing elongated and more gentle. Let's see if that's really what happens or not. So let's plug in some points. Well, notice that zero, when I square it, I still get zero, times is still zero. So, in fact, this point is still zero. What about at 1? Where originally the function was a 1, when I multiplied it by 2 it's a 2, now when I multiply it by it's only at , so in fact, it looks like our intuition may be correct. That point is sort of moved down now. Similarly, with -1 we're going to be still at . At 2, instead of being at 4 or at 8, now I'm going to be at 4 divided by 2, which is 2. So you can see it really is the case that now the parabola has been sort of elongated, and now it's more gentle. You see?
So when I multiply it by a number that's less than 1 but still positive, then it sort of makes the curve sort of less curvy, a little bit more flat in a way. When I multiply it by a factor that's actually greater than 1, then it tightens the thing up and exaggerates all the stuff.
Let's try another example. Let's take a look at g(x). Remember, I can call these things anything. I'm just going to call this g--g(x) = x^3. How does that look? We know what the graph of that looks like. That's a standard-looking cubic. Let me plot some points really fast. But hopefully, in your mind, I want you to get in the habit of thinking of a cubic, it looks like this. I'm just going to sketch it for you real fast. It comes down like this, has three wings--1, 2, 3. That's what a cubic looks like. I'm just going to plot some points to get it exact. 0^3 is 0, 1^3 is 1, 2^3 is 8, so it's way up here, and here I've got -1^3 is -1, and -2^3 is -8. So the cubic looks something like this. Let's take a look at what happens when I look at half of that. Let's take a look at the function g(x) that equals half of this cubic. What does that mean? Well, what it should do is it should make things, in fact, less steep. So somehow my guess or my thinking in my mind is, this wing's going to be coming down more like this and it will be more gradual. That's my guess. Let's see if that's really right.
So at zero I still am at zero. At 1--I used to be at 1, but now I'm going to be at 1 times , so it's . You see how the functions fall? If I put in 2, instead of being at 8, I'm now going to be at 4. And similarly down here I'm going to be a -1/2 and here I'm going to be at -4. So you can see that the cubic that I did in green is the same basic shape, but it's a little less steep, and if I were to multiply this by an even smaller number, like maybe 1/10, it would even be more dramatically less steep. So that might be something like 1/1 x^3. I'm not doing this, by the way, to scale; I'm just trying to give you a sense. Whereas, if I do something like this--if I multiplied it by a big number, let's say I multiplied by 8, that would tighten it up and it would look like this. It would be something really dramatic. So tightness means we have a big coefficient. This might be like, you know, 8x^3. A small positive coefficient will make things more relaxed. And so that's the way to think about it, and the same thing with the parabola, I remind you again. This is the parabola. If you multiply it by a big number what's going to happen is it's going to tighten up--but same basic shape. If I multiply it by smaller number, like 1/15, it sort of loosens up. It's still parabolic, but it sort of loosens up a little bit. Still a parabola, but a more relaxed parabola.
So that's what happens by stretching, and that means just multiplying the function through by one fixed number. If the number is big, the thing gets tight and more steep. If the numbers are small, but positive, then it becomes more relaxed. See, if it's big you're uptight, right? If it's smaller you're so loose--go with the flow. Try these.
Relations and Functions
Manipulating Graphs- Shifts and Stretches
Stretching a Graph Page [2 of 2]

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