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Hi, I’m Rob. Welcome to Math Antics! In this lesson, we’re gonna learn something that’s an important foundation for tons of math problems, including those you’ll encounter while learning basic Algebra. We’re gonna learn about graphing, which basically means taking mathematical relationships and turning them into pictures. Hey Friends, welcome back to The Joy of Graphing. We’re gonna pick up right where we left off. We already have this nice, beautiful function right here. But it needs a friend. And we’re gonna do that by adding some points. So let’s put the next point right here. Now all we need to do is connect those points…cuz they’re all friends. And what to friends do? They stay connected. Oh and look at that… That’s beautiful. Well, not the kind of picture that you’d hang on your wall. Graphing just means making a visual representation of an equation or data set so you can understand it better. It’s a way of helping you literally SEE how math works. When math is just a bunch of numbers and symbols on a page, it can be pretty abstract and hard to relate to. But graphing is like a window into the abstract world of math that helps us see it more clearly. In fact, the focus of our lesson today actually looks a bit like a window. …and it’s called “The Coordinate Plane”. The coordinate plane is the platform or stage that our graphing will take place on. But to understand how it works, we first need to start with its closest relative; The Number Line. You remember how a number line works, right? A number line starts at zero, and represents positive numbers as you move to the right and negative numbers as you more to the left. And there’s usually marks showing where each integer is along the way. Now imagine cloning that number line and rotating the copy counter-clockwise by 90 degrees so that the second number line is perpendicular to the first and they intersect at their zero points. What we have now is a “number plane”. It’s basically like a 2-Dimensional version of a number line, but that second dimension makes it much more useful. With a simple 1-Dimensional number line, we could show where various numbers were located along that line by drawing (or plotting) points. But no matter how many points we plot, they’re always on the same line. But with a 2-Dimensional number plane, we can plot points anywhere in that 2D area, and that opens up a whole new world of possibilities. With a 1-Dimensional number line, plotting points was easy. You just needed one number to tell you where to plot a point. But with a 2-Dimensional number plane, you actually need TWO numbers to plot each point. These two numbers are called “coordinates” because they’re the same ‘rank’ or ‘order’ and they work together to specify the locations of a point on the number plane. In fact, that’s why the number plane is often referred to as the “coordinate plane”. It’s the stage for plotting coordinates. Coordinates use a special format to help you recognize them. The two numbers are put inside parentheses with a comma between them as a separator. So when you see 2 comma 5 or negative 7 comma 3 or 0 comma 1.5 you know you’re dealing with coordinates. Okay, to understand how coordinates work, remember that our number plane is formed by combining two perpendicular number lines. From now on, we’re going to refer to each one of these number lines as an “axis”. One of the axes is horizontal (like the horizon) which means the other axis is vertical (or straight up and down). And they’re often called the “horizontal” and “vertical” axes because of that. But even more often, the axes are referred to by variable-based names. The horizontal axis is called “the X-axis” and the vertical axis is called “the Y-Axis”. Why use variable names? Well there’s two good reasons. The first is that variable names are more flexible than horizontal and vertical, which relate to a specific orientation in space that may not always be relevant. And the second reason is that each of the two coordinate numbers is actually a variable that relates to a specific position along one of the two axes of the coordinate plane. And since those variables are usually called ‘X’ and ‘Y’, it makes sense to name the two axes the same way. The first coordinate number listed will be called ‘X’ and the second coordinate number listed will be called ‘Y’. And we’re always going to list the numbers in that same order, ’X’ first and then ‘Y’, so that we never get confused about which is which. In fact, coordinates are often called “ordered pairs” because they’re a pair of numbers that are always listed in the same order: ‘X’ value first… ‘Y’ value second. So if you have the coordinates, (3, 5) that means X=3 and Y=5. Pretty easy, right? But now how do we actually plot these coordinates (or ordered pairs) on the coordinate plane? Well, the first number in the ordered pair tells you where along the X-Axis the point is located, and the second number in the ordered pair tells you where along the Y-Axis the point is located. The two numbers in an ordered pair work together to define a single point, and each one of the numbers only gives you half of the information about where that point is. To see how this works, let’s plot the coordinates (3, 2) First, we locate the X value along the X-axis, which is at 3 in this case. But instead of putting a point there, we draw (or just imagine) a line perpendicular to the X-axis that goes through the 3. We do that because the first number in the ordered pair only tells us where along the X-axis the point is, but it could be ANYWHERE along the Y-axis. We won’t know that until we plot the second number. So temporarily, we just draw a line there to represent every possible point that could have an X value of 3. Next, we locate the Y value along the Y-axis, which is at 2 in this case. But again, instead of putting a point there, we draw (or just imagine) a line perpendicular to the Y-axis that goes through the 2. We do that because the second number in the ordered pair only tells us where along the Y-axis the point is, but it could be ANYWHERE along the X-axis. So we just draw a line there to represent every possible point that could have a Y value of ‘2’ Ah, but look what we’ve got now. The first line represents all the possible locations where X equals 3. And the second line represents all the possible locations where Y equals 2. And the exact point where the two lines intersect represents the only point in the entire coordinate plane where both X=3 and Y=2. That intersections is the location of our point. Pretty cool, huh? And that’s a really good way to understand how the coordinate plane works. But I want to show you an even easier (and more intuitive) way to actually plot points. This more intuitive way involves starting with a point at the origin of the coordinate plane and then treating the coordinates like a set of simple instructions that tell you how far to move your point in the X and Y directions. For example, to plot the coordinates (3, 2) like before, we start by imagining a point at the origin (0, 0) Then, we look at the first number in our ordered pair to see how far we need to move our point in the X direction. Since X is positive 3, we move our point a distance of 3 units in the positive X direction. And then from there, since Y is positive 2, we move our point a distance of 2 units in the positive Y direction. So that’s a pretty easy method for plotting points! Let’s try it a few more times so you get the idea. Let’s plot the coordinates (-4, 3). Again, we start by imagining a point at the origin and then let the coordinates tell us how far to move it along the X and Y axes. Since X is negative 4, we move the point a distance of 4 units, but this time in the negative X direction which is to the left. And then, since Y is positive 3, we move the point a distance of 3 units in the positive Y direction. Now let’s plot the coordinates (-3, -3). In this case, X and Y are both negative, so starting with a point at (0, 0) we move it 3 units in the negative X direction, and then 3 units in the negative Y direction. And last, let’s plot the coordinates (4, -2.5). Starting at (0, 0) we move the point 4 units in the positive X direction and then 2 and 1/2 units in the negative Y direction. Okay, so we’ve plotted four ordered pairs the easy way, and did you notice that each of these points is located in a different region of the coordinate place. These four regions are called “Quadrants” and their boundaries are defined by the two axes of the coordinate plane. The quadrants are named 1 through 4 so we can easily refer to them in conversations if we need to. Quadrant 1 is the upper right quadrant, and it contains all of the points where both the X and Y values are positive. Quadrant 2 is the upper left, and it contains all of the points that have a negative X value and a positive Y value. Quadrant 3 is the lower left, and it contains all of the points that have both a negative X and a negative Y value. And Quadrant 4 is the lower right, and it contains all of the points that have a positive X value and a negative Y value. Roman Numerals are usually used to label the four quadrants and they’re in that order because that’s the order you would encounter the quadrants if you started with a line segment from the origin to the coordinate (1,0) and then rotated that line counter-clockwise around the origin. Doing this sweeps out a shape called a “unit circle” which is divided into four quadrants just like the coordinate plane. Alright, so now you know what the coordinate plane is, and you know how to plot points on it. But you might be wondering, “What has this got to do with basic Algebra?” Well, Algebra involves many different types of equations and functions that are a lot easier to understand if we graph their solutions on the coordinate plane. And as you know, the way to really get good at math is to practice what you’ve learned by doing some exercise problems. Thanks for watching Math Antics, and I’ll see ya next time. Ohh… ohhhh… [snickering] [sarcastically] Ah… exactly what I wanted. Learn more at www.mathantics.com