The Coordinate Plane
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With two axes forming the shape of a cross, the coordinate plane is used to graph algebraic, geometric, and trigonomic equations.
The axes
The two axes of the plane are referred to as the x-axis and the y-axis. The x-axis is the horizontal component and the y-axis is the vertical component. These axes go on forever—they extend infinite lengths, but, for the most part, graphs display only the numbers that are needed to show the equation correctly.
The numbering on the x-axis is done like that of a number line on which there is one central point (0) with the positive numbers to its right and negative numbers to its left. The positive numbers are equally spaced and in ascending order. The negative numbers are also equally spaced and in descending order (the lowest numbers are to the left—for example, -89 is to the left of -8 because -89 is a lower number than -8.)
The y-axis is numbered so that positive numbers are to the top of a central point and negative numbers are to the bottom. The positive numbers are equally spaced and ascend with height while the negative numbers (as you probably could guess) are also equally spaced and descend the further down you go.
When drawing a coordinate plane, many people find it easier to extend lines out from each point on each axis (like the black lines on the coordinate plane above) so there are already pre-made points and the graph will be as accurate as possible.
Origin
The very center of any coordinate plane is called the origin. This is the only spot in which the two axes meet. The central points of both axes are located here. Therefore, the only place that the two axes meet is at their central points.
Coordinates/Ordered Pairs
Coordinates are numbers that correspond to specific points on the plane. An ordered pair is made up of two coordinates: an x-coordinate (where the point is located in regards to the x-axis) and a y-coordinate (where it’s located in regards to the y-axis.)
For example, say you need to graph the point (3, 4). The first number inside the parenthesis tells us that our point can be crafted by finding the number three on the x-axis. The second number tells us that it can be crafted by finding the number four on the y-axis. One point can’t lie on both axes at the same time unless it is the origin, so this point will be located somewhere inside one of the quadrants (look below for an explanation of the quadrants.)
To graph the point that an ordered pair stands for, start with the first number. Go along the x-axis and find the point that corresponds that that number. After finding that point (the x-coordinate,) go up or down from there to whatever point on the y-axis corresponds to the second number of the ordered pair (the y-coordinate.)
When your ordered pair contains a zero as one of the coordinates it just means that your point will lie on either the x or the y-axis. For example, if my ordered pair is (0, 4), my point will lie on the y-axis because there is no value for the x-coordinate. If my point is (4, 0) my point will lie on the x-axis because there is no value of the y-coordinate.
Try to graph the point (3, 4) on the graph above. All you need to do is count three to the right of the origin and then go up four.
Getting to know the quadrants
The two axes form four quadrants which are numbered starting from the top right-hand side of the plane, over to the top left, down to the bottom left, and back over to the bottom right.
The first quadrant contains all positive numbers on both axes. Any point whose x and y-coordinates are both positive can be found on quadrant I.
The second quadrant contains all negative numbers on the x-axis and all positive numbers on the y-axis. Any point whose x-coordinate is negative and y-coordinate is positive will be in quadrant II.
The third quadrant contains all negative numbers on both axes. Any point whose x and y-coordinates are both negative will be in quadrant III.
The fourth quadrant contains all positive numbers on the x-axis and all negative numbers on the y-axis. Any point whose x-coordinate is positive and y-coordinate is negative will be in quadrant IV.
What if…?
If the equation that you’re graphing deals with large numbers, you might find it easier to make each point on each axis stand for more than just one more number higher than the point before it. So, if I need to graph (40, 75), I might make each point on each line stand for 10 points so I’d only have to go 4 across and 7.5 up instead of 40 across and 75 up (and, by default, waste a lot of time, space, energy, and lead/ink (but I suggest graphing with pencil.))
I can almost assure you that you won’t always be dealing with exact numbers; sometimes you’ll have to graph an equation with odd decimals and fractions. When this is the case, just imagine 10 (or so) really tiny points between each numbered point on the axes. This will keep you from getting thrown off-track and having a messy, inaccurate graph.
For example, to graph the coordinate (3.25, 0) just split the square between the 3 and the 4 on the x-axis up into four parts and put your dot on the first of the four parts.
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