Mapping the Plane: Coordinates and Graphs

Imagine you are a navigator on a ship with no GPS. You have a grid map and one job: pinpoint your exact position so rescuers can find you. You call out two numbers — and that is all anyone needs. Those two numbers are coordinates, and they have guided sailors, architects, and game designers for centuries. Before we dive in, warm up your memory with this question.

**The coordinate plane** A coordinate plane (also called a Cartesian plane) is a flat grid made by two number lines that cross at right angles. The horizontal line is the x-axis and the vertical line is the y-axis. The point where they cross is called the origin, written as (0, 0). Every point on the plane has an address written as an ordered pair: (x, y). The x-value tells you how far to move left or right from the origin. The y-value tells you how far to move up or down. Always read x first, then y — think 'along the corridor, then up the stairs'. **All four quadrants** The two axes divide the plane into four regions called quadrants. In Quadrant I both values are positive (e.g. (3, 5)). In Quadrant II x is negative and y is positive (e.g. (−2, 4)). In Quadrant III both values are negative (e.g. (−3, −1)). In Quadrant IV x is positive and y is negative (e.g. (4, −2)). Negative coordinates simply mean you move left (negative x) or down (negative y) from the origin. **From equation to graph** A linear equation is an equation whose graph is a straight line. The most common form is y = mx + c, where m is the gradient (steepness) and c is the y-intercept (where the line crosses the y-axis). To draw the graph, you choose at least three x-values, substitute each one into the equation to find the matching y-value, plot the (x, y) pairs, and draw a straight line through them.