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Math 101: College Algebra11 chapters | 81 lessons
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Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.
You've probably used grids before in real life. For example, a chessboard is a grid. Along the bottom, there are columns labeled A through H, and along the side, there are rows labeled 1 through 8. If you want to talk about a specific square on the board, you can refer to it by its number and letter. For example, the pawn here is at B3.
The type of grid used in math problems is called the Cartesian plane or xy plane. A Cartesian plane is a little different from a chessboard, but you can think about it the same way. Using a Cartesian plane can help you solve all kinds of math problems and understand the relationships between things in the real world.
In this lesson, you'll learn about the parts of a Cartesian plane, so you'll be all ready to use it when you need to.
On a chessboard, the horizontal location of a piece is labeled with a letter and the vertical location is labeled with a number. On the Cartesian plane, it's a little bit different: both the horizontal and vertical position are labeled with numbers. On a Cartesian plane, you can also have locations that are described with negative numbers.
The numbers that describe a particular location on the Cartesian plane are called coordinates. The x-coordinate of an object on the Cartesian plane tells you how left or right it is from the center of the graph. Every graph has a line called the x-axis, which marks the horizontal location of points on the graph. This is basically a number line, and if you've used number lines before, it should look very familiar.
If you're moving to the left along the x-axis, the number will be negative, and if you're moving to the right, it will be positive. For example, the x-coordinate of this point is 5, because it's 5 units to the right of the center of the graph. The x-coordinate of this point is -5, because it's 5 units to the left of center.
Using x-coordinates lets you distinguish between points: for example, Point A in this graph has an x-coordinate of 3, and Point B has an x-coordinate of 5. But how would you distinguish between Point B and Point C? They're obviously not in the same place, but they both have the same x-coordinate!
That's where the y-coordinates come in. The line that shows the vertical position of points on the graph is called the y-axis. The y-coordinate of an object tells you how far up or down it is on the y-axis, relative to the center of the graph. You can remember which axis is x and which is y by looking at the letters: the letter y has a long tail that goes down, so the y-axis is the axis that goes up and down.
Looking at the y-coordinates lets us describe the differences between these two points. Point B has an x-coordinate of 5 and a y-coordinate of 2. Point C has an x-coordinate of 5 and a y-coordinate of 4.
It's a little bit clunky to walk around saying things like 'This is the point with an x-coordinate of 5 and a y-coordinate of 2,' so we use a shorthand to talk about it. Instead of all that stuff, you can write the numbers as an ordered pair.
An ordered pair is defined as a set of coordinates shown as two numbers inside parentheses. The x-coordinate always comes first, and the y-coordinate always comes second. So the coordinates of Point B on the graph would be (5, 2).
Here are some examples of ordered pairs.
A point that's on one of the two axes has a 0 for the other coordinate. For example, this point is labeled (5, 0), because it's 5 units to the right of the middle of the graph, and 0 units up or down. This point is labeled (0, 3), because it's 3 units up from the middle of the graph, and 0 units to the right or left.
The point at the very middle of the graph is called the origin, and its coordinates are (0, 0), because it's 0 units away from the center of the graph in both directions.
If we draw a line on the graph, sometimes the line crosses either the x-axis or the y-axis. The point where a line crosses an axis is called an intercept. If it's the x-axis, it's the x-intercept; if it's the y-axis, it's the y-intercept. A line can have either an x-intercept, a y-intercept, or both.
If you take a step back and look at this graph, you'll see that the x and y-axes divide it into four parts.
Each of these parts is called a quadrant. To make it easy to talk about quadrants, each one gets a number, starting from the first quadrant at the top right. In the first quadrant, both x and y are positive. The top left is the second quadrant. Here, x is negative because we're left of the origin, but y is positive. The bottom left is the third quadrant, where both x and y are negative, and the bottom right is the fourth quadrant, where x is positive but y is negative.
In this lesson, you learned about the Cartesian plane, which is a type of grid that mathematicians use to show the location of lines and points. As you start doing more advanced algebra and geometry, this will help you solve all kinds of problems.
The Cartesian plane has an x-axis and a y-axis that cross each other at the origin. The x-axis is horizontal and measures distance to the left and right of the origin. The y-axis is vertical and measures distance up and down. It's basically like two number lines, with one going the usual way and another one flipped on its side to measure distance up and down.
Points on the Cartesian plane are labeled as ordered pairs, in the format (x, y), where x represents the x-coordinate and y represents the y-coordinate. If a line crosses one of the axes, the point where it crosses is called the x-intercept or y-intercept.
The Cartesian plane is divided into four quadrants, starting with the first quadrant at the top right, and moving counterclockwise.
That's a lot of terminology to learn all at once, so don't sweat it if you don't get it all immediately. As you start using Cartesian planes in actual problems, you'll pick it up really fast, and, eventually, you won't even have to think about it.
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Math 101: College Algebra11 chapters | 81 lessons