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College Algebra: Using the Cartesian System

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About this Lesson

  • Type: Video Tutorial
  • Length: 7:31
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 80 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 Basics (2 lessons, $1.98)

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.

About this Author

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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I wanted to quickly give you a real run through of plotting points on a graph where you have an x-axis and a y-axis. Well, here are some axes right here. So you can see that the horizontal one we usually call x, and the vertical one we call y, and we number them in a standard way. Where they intersect is the point that's zero on both of these number lines, and then as we march off this way these become positive values. This would be 1, 2, 3, 4, 5, and so on, and then up here we'd have 1, 2, 3, 4, 5, and if my video monitor would go on forever you would see the negative things here, -1, -2, -3, and then -1, -2, -3... And actually, the whole plane, this whole Cartesian plane, is actually broken up into four quadrants. This is called the first quadrant, where everybody's positive. The second quadrant is where the x's are negative, but the y's are still positive, and the third quadrant is a very negative space--everybody's negative there. The fourth quadrant is a little bit happier because the x's are positive and now the y's are negative. Okay, we'll talk more about that later.
And then when we plot points the convention always is to give them as a pair, the x first, and then the y, and then we'd write them in the following way. For example, suppose I wanted to plot the point of (4,5). I'd write it like this. Now, we've got to be a little, teeny bit careful, because even though mathematicians are really fastidious and really retentive, sometimes there are problems, and this is an example of one, because it's a notational issue. Do you remember how we represent interval notations. Well, if I just showed you that all by itself and covered up everything here, like that, well, what is that? Well, those could be an ordered pair for, 5 that you're going to graph on a Cartesian plane, or it could be interval notation for all the points between 4 and 5, but not including the end points. Remember all that stuff? So you've got to be really careful.
Now, it turns out, happily, mathematicians aren't that crazy, and when they give something like that; they would never do it with their arms like this. When they do it, in fact, their arms would be down here by their sides, so you always know the context in which this thing is given. And so you'll know whether that symbol means an ordered pair, first do 4, then do 5, or if it's an interval. But it's a good thing to watch out for because it will be confusing if you're not sort of up on that.
Okay, well, where would this thing go? Well, we go 4 over in the x direction and then you go 5 up. So it would be 4 over and then 5 up. And so what I do is I put a dot right around there--there's my dot. Easy as pie. Now, you can graph a whole bunch of dots at once if you wanted to. Let me show you that real fast. So suppose that you wanted to graph something, some data, some given data. So for example I thought I'd give you an actual real world example of data. I gave a mid-term a while back and I thought it would be fun to count how many students come into my office for help before the exam and then after the exam and then compare. So, in fact, I'm going to put the data right there in my box here. Let's take a look at it together, shall we?
Okay, so you can see that in the left hand column I have the days to or from the exam, and notice that the negative sign there, for example, that first data point, is (-7,0). So the -7 means it's 7 days before the actual exam, and the zero tells me how many students actually came. So -7 days, so a week before the exam, of course no one was there because no one was studying until the night before. So no one came by my office. Now, let's see, the next piece of data that I collected there was four days before the exam, and so -4, and there two people came to my office, and then you can read down the thing, two days before the exam, one person came. Notice that the day before the exam, -1, I was swamped. My office was basically filled; the floor almost gave way. I had 10 people there. Notice that even zero, that's the day I gave the exam, someone actually came in early, snuck in, and there were three people that wanted last minute help. And I noticed the next day, the first day after the exam, no one came by, that's (1,0) and you can see what happens. And then people, of course, complained about their grades, so you can see on the third day 5 people came in. I passed out the exam, they were not happy. And then the fourth day I actually have a student who has a child, and so she brought the child along and so, in fact, I counted as 1 people.
Okay, well, let's plot that and see what all that looks like if you want to sort of put that on a chart here. I'll set up some axes. Here's the y-axis. Here's the x-axis. So I have -7 over here, and now I've got to put in, between -7 and 0. This is, by the way, a great challenge if you're doing this live on the fly, as we are doing here--so you know how I do this? I say, "Well, this is -7 so that means that the middle here has to be around here, so I'm going to mark this off." Because how many will I need? I think I'll need an odd number of points here, right? The midpoint here would be like around here. -7, -6, -5, -4, -3, -2, -1. Wow! Now, if you're not impressed by that, I'm sorry.
Okay, this is now -1, -2, and so on. I'm not going to mark them all in here. And then we're going to go all the way out--if you look on the chart there--I'm going to go to 4, so I'm going to just mark them, but I want to keep the same distance just to keep our consistencies. This would be 1, 2, 3, 4. Remember, the positive x-axis represents the days after the exam, the 0 here, the y-axis represents the actual day of the exam, and the negative represents the days before the exam. Let's plot here and I'll put some numbers here. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. And let's just plot away, so you can see the list there. The first one up is -7 days, a week before the exam, no one came by. So (-7,0). So I put a dot right there.
The next one up is -4,2, so I go -4, 1, 2, 3, 4, and the 2, 1, 2, so I put a dot right there. And then I see -2. -2 I had one person. So -2 I had one person, a little dip, and then -1, the day before the exam, it looks like there's 10 people. So then I go way up here to 10. And then the day of the exam I have 3 people, and then the next day I've got nobody; nobody came around. And the next day I had nobody, so (2,0), put that point there, and then finally I pass back the exam, people are not happy, so on the third day people are complaining--1, 2, 3, 4, 5. And then on the fourth day this person came with her child, and so that's 1-1/2. So I actually go 1, and then not quite 2. I go right in the middle there, that's 1. Let me just really emphasize that and make sure you see that. That is right in between there and there and I'd right that as 1 --no, I wouldn't write it that way because that would be putting the y first. That's a great mistake that people would make, like myself, but that's wrong. What I want to do is actually make it first 4, because you put the x's first, and then 1. Plotting points on the coordinate plane--piece of cake.
Relations and Functions
Graphing Basics
Using the Cartesian Systems Page [1 of 2]

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