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College Algebra: Locate Points to Graph Equations

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About this Lesson

  • Type: Video Tutorial
  • Length: 13:39
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 147 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 Equations (2 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.

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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Throughout this course together we've been thinking about a lot of equations. We've seen equations with sometimes more than one variable, like x's and y's and so forth, where, in fact, one unknown actually leads to the value of another unknown. So, two things are actually related somehow. Maybe they're proportional, maybe you have some equation where they're both in it or something or whatever. Well, a lot of times, in fact, more times than not, having a visual image--a visual idea about that relationship between two unknowns could be extremely powerful in understanding sort of what's going on. So, this really gets into a whole realm of mathematics, which we could think of this as graphing or visual mathematics. I now want to begin by just talking about graphing equations.
What does it mean to graph an equation in the plane? So of course we have the x-axis and the y-axis. It's the x-axis here and y-axis here. What we want to do is go over some number of x values and then up or down some number of y values and we can graph things. We can start to get a picture image for what something is looking like and their relationship. So now I want to illustrate this by looking at a variety of really important types of functions. The way we'll start to graph them, just for now anyway, is just to sort of see some points and see how the relationship goes.
For example, suppose I have that y = 2x + 1. So what that means is that for this thing to be satisfied I have to find x's and y's that make this true. One way to do that is to make a little table of possibilities. So, for example what I could do here is make a little chart and put like x values here and then figure out the corresponding y values. I could pick for example some negative values, maybe like -2, -1, 0, 1, 2 and so forth--just to get some values. Now for each of these x values--I'll plug that value into here for x and find out what the associated y value would be. For example if I plug in -2 here I see -2 times 2 is -4 +1 would be -3. Here when I plug in -1 for x, I would see that 2 times -1 which would be -2 + 1 would be -1. If I plug in a 0 here, I would see just plus 1. If I plug in a 1 here, I'd see 2 + 1, which is 3. If I plug in a 2 in here, I would see 2 times 2 is 4 plus 1 is 5.
So now I have a chart of points. You can think of these as ordered pairs. (-2,-3), (-1, -1), (-0,1), (1,3), (2,5). And I can actually plot those points. So, let's try to do that right now. Here's the y-axis. Here's the x-axis. What do we see here? (-2,-3)--so let's say this is -2. That's -2 in the x -- -3 down. So I put a dot right there. That's (-2,-3). Then I have (-1,-1), (-1,-1), put a dot there. Then I have 0--so that's just on the y-axis--if x is 0 we're on the y-axis--1. Then I have (1,3). So I go 1 over and 3 up. Then I have 2 and then I go up 5. So one thing I can do is just connect those points and see what this thing visually looks like. If I connect those points--what you see is what looks like a straight line. A straight line seems to pass very well right through all those points.
So, in fact, this kind of object--where you see a x and a y all to the first power like that will represent something that graphically looks like a straight line. I want us to all start getting in the habit of looking at this and just thinking "some kind of straight line." I don't mean to know what kind--there are a lot of different straight lines. This straight line turns out to be the one associated with this. So there's our first really basic kind of function. It's a line function.
Okay, let's take a look at another one together. The next one I want to take a look at is the following. How about y = 3x^2? Again I'll make a table just to see what these values look like. As I change x what happens to y? So, I'll pick--let's say -- -2, -1, 0, 1, and 2. You can pick other points too. I'm just picking these just to illustrate what's going on. So for each of these x values I have to square it and multiply it by 3. So -2^2 is -4. No, -2^2 is actually 4 times 3 is 12, -1^2 is 1 times 3 is 3, 0^2 is 0 times 3 is 0, 1^2 is 1 times 3 is 3, 2^2 is 4 times 3 is 12. Now I have all these values and let's see happens if we plot them. If we plot them--I have -2, so, one, two in the negative directions and 12. So this point right here is the point (-2,12) and then (-1,3). So (-1,3)--look how steep that is--it really drops, really drops a lot--pretty dramatic. (0,0)--so then I go right to the origin. (1,3)--so then I go 1 over and 3 up and then (2,12) is way up here.
So, it sort of comes down and then goes up. Look how symmetric those points are. In fact, they line up perfectly like a little mirror. Sort of interesting. So what would the graph--what would it look like if you sort of connect those points? Well if we connect those points I'd have to sort of bend this thing up to fit them and it sort of looks like this doesn't it? Sort of looks like this. Actually this is an example of a parabola. In fact, whenever you see an equation like this where one of the variables just is to the first power but the other variable is a square, then you're going to have a parabola. It turns out that if this coefficient here is very large the parabola will be very, very tight. It'll be very, very tight--like this is very, very tight. If that number were smaller, the effect would be to make this thing a little bow out more. If it's really small it would even--like this and the parabola would become very wide and so forth. If that number increases, then the parabola tightens up like this and wants to close up like that. So, in fact, this is a great example of a parabola and you can see I found that just by plotting points.
Let's try another one. Let me bring this back up here because the next one is so close to this one -- I wanted you to see the difference. The next one is actually this--y = 3x^2 + 1. So, in fact, it looks like the exact same function as this, but I just added 1 to the end here. In fact, what I'll do is I'll just make my chart in black--so you know the black corresponds to the black function and this purple corresponds to the purple stuff here. Again, I'll put down -2, -1, 0, 1, 2. If I put in -2 in here and square it and multiply by 3 I get 12, but now I have to add 1, so I have 13. -1 in here produces a 1, 3 plus 1 is 4. 0 in here makes a 0 plus 1 is 1. 1 in here makes 3 plus 1 is 4 and then 2 in here makes a 12 plus 1 is 12. You'll notice that, in fact, these values are exactly the same as the y values, but they've all been increased by 1. 12 plus 1 is 13, 3 plus 1 is 4, 0 plus 1 is--and that makes sense. Because in each case all I did is take the old answer, but increased the y by 1. So each of these values are 1 more then their counterparts.
So what would happen if I graph that? If I graph that what I would see is every point would just go up by 1. You see? So this would be up by 1 unit. This would be up by 1 unit. This would be up by 1 unit--up by 1 unit and up by 1 unit. If I looked at the graph of that it would be the exact same graph that we had before--this was the graph we had before--let me remind you, but now I have to move it up by 1. Watch what happens. Exact same graph, but just shifted up by 1. So notice that it's still a parabola it's just that now it moved up a little bit because I added 1 to the very end. Without adding 1--I get that. I add 1--I get that. What do you think would happen if I put a 2 in there? That's right, it just go up by 2. See how easy that is. What if I had a -1 at the very end? Then a -1 would actually--instead of being here, I'd push down by 1. So you can see just by adding how the effect it has on the graph. So again, a very visual idea corresponding to doing the arithmetic on a function.
Let's try another example--x = y^2. Let's make a table here--x and y. Now notice something here--if I start to plug in a value--like I put in 2 here, it's going to be sort of hard to figure out what the y's are. If I put in a 3 here--okay, what's that? It would be easier to actually put the y values in and then fill in the x values. So, let me actually do that. I'll put in -2, -1, 0, 1, and 2 here. These are y values and I now want to see the corresponding x values--because it's easier. If I put in a -2 for y--if I square it I see 4. -1^2 is 1, 0^2[ ]is 0, 1^2 1, 2^2 is 4. So now I have these points. Let's see what they look like. Here's the axis and let's graph. So I have 4 in the x and I go -2 in the y. So, I put a point right there. Then I have (1,-1). So I put a point right here. Then I have (0,0), so I put a point right there. Then I have (1,1) so I put a point right there. Then I have (4,2). So, I put a point right there. If we connect those things what do we see? Well we actually see a curve that looks sort of like this. Again we see a shape that looks like a parabola--this parabola curve.
Notice something interesting--in that previous example where the parabola went like this--notice that was the case where we had the x squared. Now we have the y squared and notice the parabola is going in this direction. That's not a fluke. In fact, if you have y squared the parabola will either be like this or like this. Whereas if we have x squared and the y is just appearing to the first power the parabola will be either up or down. So parabolas with y squared will look sort of like this or like this and with x squared the parabolas will look like this or like this. So that's sort of neat. In this case this parabola looks just like that--sort of a sideways parabola.
I thought I'd do one last one with you and this last one is y = |x|. Let's make a table of points here--x and y and see what happens. -2, -1, 0, 1, 2. So we're just taking the absolute value of x--so absolute value of -2--that's just 2. Absolute value of -1 is one. Absolute value of 0 is 0. Absolute value of 1 is 1, 2 is 2. That was pretty fast. What does this look like? -2 and I go up 2 -- (-1,1) and (0,0). Looks like a nice straight line doesn't it? Well let's see what happens now. (1,1)--uh-oh--so much for the straight line. (2,2)--so it looks like almost sort of a parabola like thing, but, in fact, it's not. It goes straight down like this and then comes straight out. So, in fact, the look is something like this. It comes straight down and then goes straight out. So it makes this sort of sharp V here and this is what the absolute value function looks like. It usually is made up of two wings and they're both very sharp like this. One goes straight down. One goes straight up like that.
These are some very, very basic functions where we're getting a sense of what the picture of what these relationships look like by making a little table and graphing. Up next we'll start learning how to graph things in a slightly more sophisticated way. For now, visual with each picture, with each equation.
Relations and Functions
Graphing Equations
Graphing Equations by Locating Points Page [2 of 2]

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