A Map for Numbers
How can you describe a specific location on a flat surface, like a map or a screen? You can use a coordinate plane (also called a Cartesian plane). It's a two-dimensional grid formed by two number lines that intersect at a right angle.
The Parts of the Coordinate Plane
x-axis: The horizontal number line. Positive numbers are to the right, and negative numbers are to the left.
y-axis: The vertical number line. Positive numbers are up, and negative numbers are down.
Origin: The point where the x-axis and y-axis intersect. Its location is (0, 0).
Quadrants: The two axes divide the plane into four sections called quadrants, numbered with Roman numerals starting from the top right and moving counter-clockwise (I, II, III, IV).
Plotting Points
We can identify any point on the plane using an ordered pair of numbers, written as (x, y).
The first number is the x-coordinate. It tells you how far to move horizontally from the origin (right for positive, left for negative).
The second number is the y-coordinate. It tells you how far to move vertically from the origin (up for positive, down for negative).
The rule is always 'run before you jump'—move along the x-axis first, then the y-axis.
Example: To plot the point (-3, 4):
1.Start at the origin (0, 0).
2.Move 3 units to the left along the x-axis.
3.From there, move 4 units up parallel to the y-axis.
4.Mark the point. This point is in Quadrant II.
Graphing Linear Equations
The coordinate plane allows us to visualize equations. A linear equation is an equation whose graph is a straight line. The most common form of a linear equation is the slope-intercept form:
y = mx + b
b is the y-intercept. This is the point where the line crosses the y-axis. Its coordinate is (0, b). This is your starting point for graphing.
m is the slope. The slope tells you the 'steepness' of the line. It is often written as a fraction, 'rise over run'.
Rise: How many units to move up (positive) or down (negative).
Run: How many units to move to the right.
Example: Graph the equation y = 2x - 1
1.Identify the y-intercept (b): b = -1. So, the first point is at (0, -1). Plot this point.
2.Identify the slope (m): m = 2. It's helpful to write this as a fraction: 2/1. This means Rise = 2, Run = 1.
3.Plot more points: Starting from your y-intercept (0, -1), 'rise' 2 units (go up 2) and 'run' 1 unit (go right 1). Plot a new point at (1, 1).
4.You can do this again from your new point: rise 2, run 1. Plot a point at (2, 3).
5.Draw the line: Use a ruler to connect the points with a straight line.