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College Algebra: Matching Equations with Graphs


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About this Lesson

  • Type: Video Tutorial
  • Length: 11:54
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 127 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: Graphing 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 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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So, I thought it would be interesting to actually take a look at a whole bunch of different functions written in sort of algebraic form and try to get a visual image to what the graph looks like. So, that' right, it's time for that great, late night game show. The game show that's sweeping the country, it's time for "What's My Curve?" with your host, of course, me, Ed Berger. So this is going to be a course where I give you the equations of many functions and then, one at a time, show you a graph and your job is to determine which equation corresponds to that graph.
Let's meet our contestants. Shall we? First we have, coming from Dubuque, y = the square root of x. From Chicago we have x = y^2. From New York City we have y = ^1/[x], of course. From Kansas we have y = [x]. From Nebraska we have y = |x|. From Detroit we have y = x^3. From New Jersey we have y = x^2. Coming all the way from Florida, ladies and gentlemen, we have x = |y| and finally, right here from Austin, Texas we have y = the cube root of x. Let's welcome our contestants. So, everyone knows how this game is played. I show you the graph of a function and you're job is to figure out which of these, if any, correspond to that graph. Let's start. Shall we?
Here's our first graph. Now your job is to figure out which of these, if any, actually corresponds to this graph. Give it a try right now. This is an interactive moment.
Well, actually, it looks like we start of with a trick question right away. I believe that none of these actually give rise to this. Because this is a straight line function, so it should have a y = mx + b flavor and I see that the slope looks like it's one, since it goes through the origin at (1,1). The y intercept is 0, so this looks like the good old favorite, every popular, much beloved, y = x. We had her on last week, but apparently not on today's show. So, this answer is none of the above, or in this case, none of the to the left.
Okay, let's try another example. This is our second graph, graph number two. We'll give you all a moment at home to try this. See if you can find the right function.
Well, one thing you can do is just try to plot some of these points and see if any of them correspond. Let's see, at 1 we're 1 and at 4 we're 2. If you look at that for a second you may see that it sort of looks like part of a parabola. In fact, it is half of a parabola, but it's half a parabola going this direction. So that means that probably this is going to be something with y^2 in it. So, I'm looking for something with y^2 in it and here's one with y^2. Let's see if that's actually going to be correct or not. Well, if I put in for example, a x = 4, what do I see? If I put in x = 4, I see that--well, y would be either + or - 2. So, that's not good, because I only have a +2 here. I don't have the -2. So, this is probably not going to be the right answer. What I want is, I want that half the curve that top half. That's the positive square root.
So, what if I took square root of both sides, but only kept the positive square root since I don't want negative part of the parabola? Well, then I would have y = the square root of x. Well, that's contestant number one, ladies and gentlemen, right there. So, the graph of this is y = the square root of x. Why? Because this is always positive and this is the square root. Don't be fooled, as many contestants are, by a square root being + or -. It's just positive, unless there's + or - in front, and you can check. If you plug in 1, square root of 1 is 1. If you plug in 4, the square root of 4 is 2. This is the graph of y = the square root of x. Notice it is half of the parabola that you see over here. So, that is the answer. Congratulations those of you who got it right and those of you who got it incorrect, don't worry, there's more to come.
Graph number three. Here we go. What does that look like? Try it now.
Well, let's see. I see this sort of curvy, kink in it. So, it seems to me if I were to look at it this way, for example, it looks sort of like the cube function we saw earlier. So, this might be some sort of cubic function. Now, let's see if we can figure out exactly what this is. Notice that if I put in an 8, I get a 2. That means I'm somehow taking maybe the cube root. Is there any cube root functions here anywhere? There's one right here, the cube root of x. Let's see if this check or not. If I put in 1 the cube root of 1 is 1 and that corresponds here. If I put in 8, the cube root of 8 is 2. If I put in -1, can I take the cube root of -1? Absolutely, because it's an odd root. The cube root of -1 is -1. Don't believe me? Take -1 and cube it. -1 times -1 times -1 is -1. Great. What about -8? Cube root is -2. Ladies and gentlemen, this is the cube root function, y = cube root of x. Congratulations to all those winners out there in web land.
Our next function looks like this. A little bit exotic. It's the famous V function that you may have seen on TV. Let's see if you can guess the appropriate function here. Good luck.
Well, when I see a V function like this I think absolute value, because I think of two wings coming together. But this, by the way, is not a function of x. It's maybe a function of y, but it dramatically fails the vertical line test, as you can see, all over the place. So, I'm not looking for a y = something. I'm looking for a x = something. Let's see if there's any x = somethings with square roots. Doesn't look good. Oh, but we're saved by the very, very last person right here, from I think, Austin, Texas or somewhere.
Anyway, it's x = the square root of y and that's exactly right. Look what happens if you try to plug something in. If I try to plug in, for example, 2--if I put 2 in for x, what are the possible values for y? The possible values for y are 2 and -2. Both those values have absolute value that equals 2. So, this is the correct answer. Notice it is not a function, because I could not solve this for y without having two possible answers. So, this is not a function. It is a function with respect to y, but not with respect to x. So, I have x = stuff. So, x = |y|. There we go.
Let's try another one real fast. We have this happy face curve. See if you can guess the right equation for this.
Well, this is the ever popular and much beloved parabola. There is no discussion needed, it's y = x^2. It's centered at the origin. If you noticed, 2^2 is 4, -2^2 is 4. This is just a parabola. Wherever you see y = x^2, this picture should be etched in your mind. Congratulations to everyone who got that correct.
Let's see what else we have here. The quite exotic one. This is coming in from San Francisco and you can see this one is really complicated. A lot jumps here. See if you can guess which of these remaining functions corresponds to this and good luck.
Well, of course it's the greatest integer function, because this function only takes on integer values. It jumps as you jump from integer to integer to the next integer value. This is the standard y = [x] function.
Good luck on this next one. Well, the choices are limiting here, so by the process of elimination, the chances are getting higher and higher that you'll get some of these correct--for those of you that are just guessing randomly. Anyway, look at this one. First of all, ask yourself, "Is it really a function of x?" That may be a clue.
Well, it's not a function of x. It dramatically fails the vertical line test. So, I should have an x = something in terms of y. What should it be? Well it does look like a parabola on its side. So, I should be looking for a parabola in terms of y. There it is, our number one candidate right there, x = y^2. You can check that if you want. For example, if you put x = 4, what are all the y values that when I square it I get 4?" Well, it's 2 and -2. You can see, in fact, this is a sideways parabola. Whenever you see x = y^2, you should be thinking of a sideways type parabola. Okay. Great.
We're coming to the end of our show here. I hope you've had fun, by the way ladies and gentlemen. Here's one to try. Looks pretty exotic. See if you can guess the appropriate function for this.
Well, it's the y = x^3 function and that's easily seen by plugging in a 2 and seeing that 2^3 is 8. Plugging in a 1 and see 1^3 is 1. Similarly with the negative values, -2^3 is -8. So, you can see this thing and this is the standard form of a cubic. Notice that it's a cubic--it has three little wings. I wonder if one of my assistants can give me back the quadratic, which is somewhere down there. You'll notice that a cubic has three wings. It doesn't look like it, but you see there's a wing here, there's sort of a wing and then it bends and then it bends again. There are three bends and it's a cubic. Now, look at the quadratic.
The quadratic is y = x^2 and notice it has two bends. It sort of bends here and then bends and you go up here. So, we have one, two wings and it's a quadratic. If we have a cubic, we have three wings. So, this is a standard form of this standard cubic. Always good to tell how to find a cubic by how many little bends there are.
Our penultimate graph is this. Try your luck at this one.
Well, it's certainly a function. I see that classic V. That classic V reminds me of absolute value, because whatever I do on the right, I'm going to do the same thing on the left. Same values, just strip away the negative signs. So, I see this line and that line together, form y = |x|. Well that leaves one last one. Let's see how we do here. Well, ladies and gentlemen, it does correspond to y = ^1/[x]. If you think about that, the way we can see that is the following. Notice that if we plug in some points here, for example, 1, ^1/[1] is 1. If I plug in ^1/[2, ]1 over ^1/[2], that's actually a compound, complex fraction. If I invert and multiply, what I see is I get 2. Because 1 over ^1/[2], 1 flipped is just 2. At 2, I plug that in, I get a half.
So, I get this wing of the--and this is called a hyperbola. Then this in another wing of a hyperbola. For example, if I plug in -2, I get (2,-^1/[2]). So, this hyperbola is quite exotic. You see that it actually never crosses the y-axis because there is no value of x, which will make this 0. Notice that I'm not allowed to plug in 0 here, because if I put in 0 here--in fact, this is undefined. It never crosses the x-axis for the very reason I just said, there's no value of x that will allow me to actually have y = 0.
Well, I hope you did fine on your game show. I'm sure you did. Even if you got wrong answers, the important thing was playing the game at all. We have some lovely parting gifts, so enjoy these and now here's a word from our commercial sponsors.
Relations and Functions
Graphing Functions
Matching Equations With Their Graphs Page [3 of 3]

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