The first thing you got to know is that there are four quadrants in a coordinate plane. What’s wacky is how these quadrants are arranged:
A good way to think of it is to imagine yourself on a cross-country trip in which you start off in New York (the northeast = quadrant I), drive on over to Seattle (the northwest = quadrant II), then drive down to Los Angeles (southwest = quadrant III) and finally end up in Miami (southeast = quadrant IV).
The next two important things you have to learn are the x-axis and the y-axis. The x-axis runs from left to right (this is the horizontal line), whereas the y-axis runs up and down (this is the vertical line).
Next, it’s all about integers, or what I call “people numbers”: you can have one more person or one person fewer. However, you can’t have .3 people or ½ a person. In coordinate geometry land you’ll have integers running from left to right, like on a number line, from negative to positive. On the left hand side the negative numbers will decrease until they hit the middle of the graph (the intersection of the x- and y-axis—of what’s know as the origin). At that point, you’ll see the number zero. Then the numbers continue up from zero (1, 2, 3, etc.).
For the y-axis, the vertical one, the negative numbers start at the bottom, decreasing until they get to zero. Then, from the origin going up, each number will go up by one for each “notch” in the graph.
Next up are the coordinates. Coordinates are always presented as two numbers, the first number corresponding to the x-axis, and the second one to the y-axis. For instance, point A has the coordinates (2, 4). That is you “go over” two points to the right on the x-axis and up four points on the y-axis. Point B has the coordinates (-1, -3).
Even though the numbers on the x-axis and y-axis are always broken down via integers, coordinates can have fractions, as point C shows (1/2, -7/2).
Now that we’ve gone over the “basic basics”, next up is something that might sound familiar to you: rise over run. You’ve probably had this hammered in for so long you can’t even remember when you first heard it. Sadly, most people forget exactly what it means. So here it is: you can figure out the slope of a line by seeing how many squares on the coordinate plane one point is above the other. In the graph below, point D has a y-coordinate of 3 and point E has a y-coordinated of -2. So point D is five squares higher than point E. This is the “rise”, how far up it goes from the lowest point, is 5.
When you are figuring out the slope, which consists of a fraction, always put the “rise” in the numerator.
As for the “run”, “it” is how far away two points are in a left to right sense. If you count how many squares point D (3, 2) is from point E (-2, -3), in a left to right sense, you’ll get the “run”, which in this case is 5. I know, you are probably thinking, why the heck is that called the “run”. You can run from left to right, right to left, up to down, and just about everything in between. It’s just one of those conventions you have to learn. The key is you always put the “run”, or whatever you want to call it, in the denominator.
You might have already figured out that there is a faster way of doing this besides counting squares. All you have to do is take the y-coordinate from point E and subtract it from point D, which gives you (2-(-3)) = 5. And to find the “run” just subtract the x-coordinate of point E from the x-coordinate of point D, which gives you (3-(-2)) = 5, so the slope is 5/5 = 1.
Now as far as the slope goes, I’ve left something out that would make your geometry teacher’s face turn red. y(subscript)2 – y(subscript1)/x(subscript2) – x (subscript1).
As you can tell, this is pretty ugly. It’s useful for sure, but it’s just not that most straightforward way of dealing with the slope. For one, students usually get flustered trying to remember which one is the y(subscript2) and which one the y(subscript1). It actually doesn’t matter—as long as you make sure to start with the same point for both coordinates. So above I have the y-coordinate in D – the x coordinate in E. Therefore, I have to make sure that I have the x-coordinate in D – the x-coordinate in E.
It’s also okay to switch the points D and E around, as long as you do it for both coordinates. You will end up getting the same thing.