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College Algebra: Exponent Function Graphs-Patterns

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  • Type: Video Tutorial
  • Length: 8:56
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 96 MB
  • Posted: 06/27/2009

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
Algebra: Exponential and Logarithmic Functions (36 lessons, $49.50)
College Algebra: Exponential Functions (3 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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So let's take a look at what the exponential functions would look like graphically. Let's start off with a really simple one. How about f(x) = 2^x? Well, the best way to get a sense of what the graph of this looks like is just to plot some points, so let's just plot some points and see what happens. So here's my axis. I guess I can make a little table over here, and I can put in x's and f(x)'s. So let's start off with zero. If I plug in zero for x--now, remember, these are exponent values. We have to be careful. This is now 2^0, which we talked about and saw equals 1. We're going to put a 1 in here. And then 2^1 is just 2. What if I put a 2 in there? Then I see 2², which should be 4. If I put a 3 in here, I see 2³, which is 8. So let's graph this. Zero, we have 1. At 1 I have 2, so 1, I go up to 2. At 2 I'm at 4, so at 2 I double. So we're way up here. And at 3 I even double this, so I'm almost off the page. Dramatic growth--just goes right up there.
Well, what about -x's? Well, where can I put them? I'll make a chart over here. This illustrates that charts can be any way at all. You don't always have to be vertical; you can be horizontal. So suppose I put in -1? Well, what is 2^-1? Remember, a negative exponent means a flip, so 2^-1 is the same thing as , which is just or . So at -1 I'm at . What about -2? Well, once you remember the negative sign flips, I see 2^-2, which is or . So look what's happening here. I'm getting smaller and smaller fractions here. So the graph, if I connect these points, seems to look like this. And that, in fact, is standard graph for an exponential function of the sort.
Now, let's think about it for a second. First of all, it seems like I never touch this line here, the x-axis. Is that true? Well, for me to cross the x-axis, that would mean that I would have a zero of this function. Is there any value of x I can plug in that would make this zero? Well, 2 to any power will never give me zero. I can't take 2 and raise it to some power and get zero. You may say, "What about zero?" No, no--2^0 remember is 1. But as I put in smaller and smaller numbers, -1,000, -1,000,000, I'm going to get 1 over bigger and bigger quantities, so I approach zero, but I never touch it. In fact, the x-axis is a horizontal asymptote off on the left. It's a left horizontal asymptote, because this curve's going to approach it but never touch it.
The other intriguing thing that's worth noticing is that with this simple exponential function, the function never seems to be negative. Does that make sense? It sure does, because whatever you raise 2 to, you can never make that thing negative; 2 to any power will always be positive--always be positive, never be negative, never be zero. So, in fact, this should always live above the x-axis, and in fact, this is the curve. Dramatic growth on the right, and then very, very gentle decrease to the x-axis, approaching the x-axis, on the left.
Now, once you know this is the general shape of an exponential, you can now ask, "Okay, what happens, for example, if I change this number?" Like what happens if I look at this--g(x) = 3^x? So what if I change this and enlarge the base, how does it affect the picture? Well, if you think about it you can plot some points. I'll let you try to plot the points. At zero we'd still be zero, because 3^0 would still be zero. But already at 1 I'll see a difference because I'll see 3^1, which is 3, so I'll be a little bit bigger. And at 2 I'd see 3², which is going to be 9, so I get bigger. And what happens is, what you see basically is that this thing takes off even faster. It lives above this line. It lives above it. What happens here? You may think, "Oh, it's going to live above it here," but actually, no, it's going to actually live below it here, and this is going to shrink faster. Right? Think, for example, what happens at -2. If I plug in -2 I have 3^-2. That's , that's 1/9, whereas when I plugged in a -2 in here, I was only at . So this is dropping really fast. So what happens is to the right I am just growing much faster; I live above it, I come down and meet right at (0,1), but then I come down below it and I undercut the competition--still asymptotic.
So, in fact, the higher you make the base here, the faster it takes off on the right. It just really takes off really fast so it's going to be above it, but then on the left it's going to jump down even faster, so it's going to come down below it, like that. Just undercut it. So that's the general shapes of these things.
So that's what happens if you increase the exponent and decrease it. I'm going to draw a picture of that to make sure you really see it. I'm going to draw a real dramatic picture just so you can really get a sense of this. So this might be like 2^x. And now, if I look at 3^x it'll be more dramatic here, above, comes to the same point, but then drops below, so that would be 3^x. What do you think 5^x would look like? Roughly speaking, where would that live? Well, that would be even higher here and then come down even lower yet, so it would be more dramatic.
Again, the higher you make it, the more drama. But it doesn't touch, it's just asymptotic. And you can plot points and see what those values are. Okay, neat. Now, what about if I look at something like this? What about if I look at 1/2^x? So what if I put in a base that's actually less than 1? Less than 1, but positive. Well, we can plot points and see, or we could just use the facts we know about functions and exponents, and realize that that actually equals 2^-x by laws of exponents. If I flip, that's a negative exponent. So I could write it as 2^-x, and what does -x do when I have a function x like this? Well, you may remember from the graphing things, that actually is a flip this way. It's a flip along the y-axis, and why? Because I take every x value and I replace it by an -x value. So I just flip the picture.
So, in fact, fractional things of this sort, , well, that picture actually is just this. It's this old picture, 2^x, flipped over. I will attempt to draw that now in brown. This would be 1/2^x and the reason why I saw that is because it equals 2^-x. 2^-x is just 2^x, but flipped along this axis so it looks like that. So, in fact, graphing things to the form 1/5^x, you just graph 5^x and flip it over here so we have a very sharp thing and drop down here.
Let's take a look up next at what happens if I put in a negative number in here, and then take a look at some slightly more exotic functions and their graphs, and we'll just be using shifting techniques there, so first I'll graph one of these simple things and start shifting all over the place. Okay. I'll catch up with you there.
Exponential and Logarithmic Functions
Exponential Functions
Graphing Exponential Functions: Useful Patterns Page [2 of 2]

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