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College Algebra: Shifting Curves along Axes


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About this Lesson

  • Type: Video Tutorial
  • Length: 4:17
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 45 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 at 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.

About this Author

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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 or visit Thinkwell's Video Lesson Store at

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You know, a lot of times in life you want to actually graph some really complicated function, and you just don't even know what it looks like, but what you may notice is that hidden inside that complicated function is actually a very, very simple function that has been sort of massaged and sort of strangled around a little bit in a form, and it may be this exotic one. If you can actually find that essence, that very core of the simple function, and if you recognize it, sometimes just by fondling that simple function you can make it out to look like the more exotic one. And that's what we're going to start thinking about now, and I wanted to sort of give you a sense of a general theme of what's ahead.
So if you have a function, let's say this function right here, sort of a complicated-looking function, there are a lot of functions that are actually related to this function that aren't any more difficult than this one. For example, what I could do is I could take this function and shift it up--and watch what this looks like. Well, that's another function. But it's just the same function, I just moved everything up a little bit. Look at this function here. Well, that's the same function, I just shifted it over a little bit and so on. So basically, once you have one function you can make a whole bunch of other functions just by shifting it around.
Now, how does that shifting actually go? Well, that's sort of an interesting question and an important one, and not too typical. Let's take a look at this within the example. So here's a nice, healthy parabola, and maybe this is the parabola that looks like y = x^2. So then you would have this thing looking like this. Okay, well, let's think about it. What would I have to do to this parabola--this is y = x^2--so this is f(x) = x^2--now, how would I modify this equation so that I could just shift it up one unit? What's the equation for the graph of this where it's just the same as the old thing, but shifted up one unit? Well, let's think about it. What does a shifting do? A shifting just increases the y values. I want to raise up, that means what I want to do is actually increase all the y values. If I want to shift it up one unit, it means that I should increase all the y values by 1.
Well, let's see, what are the y values? Well, they're given by the x^2. So, in fact, if I were just to tack on a +1 right here, that should raise this curve up one unit. Similarly, if I were to take this thing and just subtract 1, what should that do? That should take all of those y values I had before--this thing--and reduce each one by 1, lower the value. So that should just shift it down.
So, in fact, by taking a function, like x^2, and adding a number to it, what that's going to do is raise that function up, raise that curve up. If I were to subtract a number from that function, like x^2 - 7, that would take that curve and lower it down, bring it way down. So that is actually an easy way of seeing what happens when you add or subtract a number to the end of a function.
But the question still remains, how do you get the function to shift right and left? Well, it turns out that that's a little bit more tricky and we'll have to think about that together, but the answer is going to be to replace the x by x minus a number, and to basically shift the x. We'll see how that works actually in practice up in our next real example, but there will be other techniques we can use, for example, looking at symmetry, and for example, symmetry, this can be seen by taking a mirror and noticing that some graphs have the feature that if you actually look at the graph in the mirror you see the exact same graph as you would see without the mirror. See? Without the mirror, with the mirror. Without the mirror, with the mirror. I can do this, by the way, for many, many hours, but I won't.
Anyway, we'll take a look at symmetry, and all sorts of symmetry, even some symmetries, by the way, that cannot take place in a mirror. What could they possibly be? Well, first we'll take a look at how to shift these functions around in space and look at their graphs. I'll see you there.
Relations and Functions
Manipulating Graphs- Shifts and Stretches
Shifting Curves Along Axes Page [1 of 1]

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