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Hi I'm Rob welcome to math antics in this lesson we're going to learn something that's an important foundation for tons of math problems including those you'll encounter while learning basic algebra we're going to 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 going to 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 do 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 0 and represents positive numbers as you move to the right and negative numbers as you move to the left and there is usually marks showing where each integer is along the way now imagine cloning that number line and rotating the copy counterclockwise 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 two-dimensional version of a number line but that second dimension makes it much more useful with a simple one-dimensional number line we could show where various numbers were located along that line by drawing or plotting points no matter how many points we plot they're always on the same line but with the two-dimensional number plane we can plot points anywhere in that 2d area and that opens up a whole new world of possibilities with the one-dimensional number line plotting points was easy you just needed one number to tell you where to plot a point but with the two-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 location 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 base 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 specific orientations 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 it's 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 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 comma 5 that means x equals 3 and y equals 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 comma 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

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