The xy-Coordinate Plane
Coordinate geometry tests the same material as plane geometry, just on an xy-coordinate plane. Think of the xy-coordinate plane as a map of sorts: the x-axis runs left to right, similar to how east and west work on a map, and the y-axis runs up and down, or north and south. Both axes work like number lines in positive and negative directions, stretching infinitely in both directions. Ordered pairs (x, y) indicate
where on the map to plot points, the first number always referring the x-axis and the second always referring to the y-axis. The x- and y-axes cross one another at the origin, point (0, 0), and all other points are in reference to the origin. Everything to the right of the origin has a positive x-value while everything to the left has a negative x-value.
Points above the x-axis have a positive y-value, while points below the x-axis have a negative y-value. Think of this concept as the equator splitting the earth in two.
Look at the following figure, for example. The point (4, 5) indicates to travel from the origin four in the positive direction on the x-axis, and then five in the positive direction on the y-axis. Similarly, point (–4, 5) travels in the negative direction on the x-axis, but in the positive direction on the y-axis.
Try some on your own! Plot the following points: (1, 7), (–1, 7), (1, –7)
What kind of shape do these points create if you connect them? If you answered triangle, you are correct. Connecting points to create shapes can very easily turn a coordinate geometry question into a plane geometry question.
Formula of a Line
Coordinate planes are also useful for plotting lines, shapes, and curves.
For these tests, it is especially useful to understand how the formula of a line works. The most common form of the line formula is called slope-intercept form because it shows the slope and the y-intercept:
y = mx + b
The x and y in the formula represent the two parts of an ordered pair.
m represents the slope and shows how steep or shallow the line’s incline or decline is. A positive slope means that the line will rise from left to right and a negative slope will descend from left to right. To find m, you will need two points along the line to find the rise over run. Any two points define a line by connecting them together, so it does not matter which you use. The “rise” refers to the change in the y-axis, and the “run” to the change in the x-axis. Use the following formula to calculate the slope and be sure to remain consistent as to which point you call the first point and which you call the second as long as you stay consistent on the numerator and denominator.
Let’s try an example:
Points (2, 6) and (–1, 0) lie on a certain line. What is its slope?
(A) –4 (B) –2 (C) 2 (D) 4
Here’s How to Crack It
Use the slope formula using the (x, y) ordered pairs. Let’s call (2, 6) the first point, (x1, y1), and (–1, 0) the second, or (x2, y2). Therefore, the slope equation will read . Simplify the numerator and the denominator to find that . This slope will be an upward slope. The correct answer is (C).
The other important component to slope intercept form is the y- intercept, the y-value when x is zero. You can find this point if you know the slope and any point. In the previous example, the slope is 2, and you already know two points. The formula should read y = 2x + b, so to solve for b, choose a point to plug in for x and y. Try using (2, 6), for example. The equation will become 6 = 2(2) + b. Multiply 2 by 2 to find that 6 = 4 + b. Then, subtract 4 from both sides to isolate b. Since b = 2, the point (0, 2) must also be a point on the same line. The final equation to the line containing all these points is y = 2x + 2.
Parallel and Perpendicular Lines—Middle and Upper Levels Only
Lines parallel to each other will have the same slope and different y- intercepts, meaning that they will travel in the same direction and never intersect. Lines that are perpendicular to one another intersect at a 90° angle. Therefore, their slopes will be the negative reciprocals of one another. For example, if the slope of one of the lines is 2, the slope of the line perpendicular would be . If the slope were , the slope of the line perpendicular would be . Try a sample question.
The line y = 3x + 2 contains no solutions with which of the following lines?
(A) y = 3x + 2 (B)y = –3x – 2
(C)
(D)y = 3x + 3 Here’s How to Crack It
The test-makers sometimes use fancy language, but a “solution” is the same as an intersection. If the lines never intersect, that must mean they are parallel to one another, and therefore have the same slope.
Eliminate (B) and (C). For the lines to be parallel, they cannot be the same line, which would have the same y-intercept. Therefore, eliminate (A) and the correct answer is (D).