LESSON 19

THE COORDINATE PLANE

Mapping Numbers in Two Dimensions

A Grid for the Universe of Numbers

The coordinate plane is a flat surface formed by two number lines that cross at right angles. The horizontal line is called the x-axis and the vertical line is called the y-axis. Their intersection point is called the origin, labeled (0, 0). Every point on the plane can be described exactly using an ordered pair (x, y).

This system was invented by the French mathematician René Descartes in the 17th century — which is why it is also called the Cartesian plane. It bridges algebra and geometry, allowing equations to be drawn as graphs and geometric shapes to be described with equations. It is the foundation for nearly all of higher mathematics.

Ordered Pairs and Plotting Points

Every point on the coordinate plane is identified by an ordered pair (x, y). The first number (x) tells you how far to move left or right from the origin along the x-axis. The second number (y) tells you how far to move up or down along the y-axis. Positive x moves right; negative x moves left. Positive y moves up; negative y moves down. Always move horizontally first, then vertically.

Example:Plot (3, 5): start at origin → move 3 right → move 5 up Plot (−2, 4): start at origin → move 2 left → move 4 up Plot (4, −3): start at origin → move 4 right → move 3 down Plot (0, 0): the origin — no movement needed

The Four Quadrants

The x-axis and y-axis divide the coordinate plane into four regions called quadrants, numbered I through IV counterclockwise starting from the upper right. Quadrant I: x positive, y positive (+, +). Quadrant II: x negative, y positive (−, +). Quadrant III: x negative, y negative (−, −). Quadrant IV: x positive, y negative (+, −). Points on the axes themselves are not in any quadrant.

Example:Quadrant I: (3, 7) — upper right Quadrant II: (−5, 2) — upper left Quadrant III: (−4, −6) — lower left Quadrant IV: (8, −1) — lower right

Distance Between Two Points

The distance between two points (x₁, y₁) and (x₂, y₂) is found using the distance formula, which comes directly from the Pythagorean theorem: d = √[(x₂ − x₁)² + (y₂ − y₁)²]. The horizontal difference and vertical difference form the legs of a right triangle, and the distance is the hypotenuse. The midpoint between two points is found by averaging the coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Example:Points A(1, 2) and B(4, 6): Distance: √[(4−1)² + (6−2)²] = √[9 + 16] = √25 = 5 units Midpoint: ((1+4)/2, (2+6)/2) = (2.5, 4)

Graphing on the Coordinate Plane

The coordinate plane is used to graph equations. A linear equation like y = 2x + 1 produces a straight line — every point (x, y) that satisfies the equation lies on that line. To graph it, pick two or three x-values, calculate the corresponding y-values, plot the points, and draw a line through them. The x-intercept is where the line crosses the x-axis (y = 0); the y-intercept is where it crosses the y-axis (x = 0).

Example:Graph y = 2x + 1: x = 0 → y = 1 → point (0, 1) x = 1 → y = 3 → point (1, 3) x = −1 → y = −1 → point (−1, −1) Plot all three and draw the line

Key Facts

René Descartes reportedly invented the coordinate system while lying in bed watching a fly on the ceiling — he realized he could describe the fly's position using two numbers measuring its distance from two walls.

GPS technology uses a three-dimensional version of the coordinate system — latitude, longitude, and altitude — to pinpoint any location on Earth to within a few meters.

Computer screens use a coordinate system where (0, 0) is the top-left corner, x increases to the right, and y increases downward — the opposite of the standard math convention.

The coordinate plane is the bridge between algebra and geometry. Descartes' invention made it possible to solve geometric problems with algebra and to visualize algebraic equations as shapes.

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