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How to Draw a Cartesian Plane

Early in the 17th century, Rene Descartes was lying in bed one morning watching a fly walking across the ceiling. Descartes thought that if the edges of the ceiling were calibrated like the marking on a ruler, the path of the fly could be described by a changing set of numbers. From these idle musings came the invention of analytic geometry, a marriage of algebra and geometry. This in turn led to the development of calculus in the next century. The background for analytic geometry is the Cartesian plane, named in honor of Rene Descartes.

Instructions

    • 1

      Draw the Cartesian plane so that each point in the plane has an address. The center is called the origin, and it is usually drawn in the center of the page. A horizontal line goes through the origin and is calibrated to the right of the origin with positive numbers and to the left of the origin with negative numbers. The address of the origin is (0, 0) and both the positive numbers and the negative increase in size as you go away from the origin. The size of the calibrations can be whatever is appropriate for a particular problem.

    • 2

      Construct a vertical line through the origin and perpendicular to the horizontal line. Calibrate it going up with positive numbers and down with negative numbers. The vertical line is called the Y axis and the horizontal line is called the X axis. You can find the address of the point in the plane by drawing a vertical line to the X axis to get the first part of the address, and draw a horizontal line to the Y axis to get the last part of the address. The address of a point is always a pair of numbers enclosed in parentheses.

    • 3

      Use the plane for graphing functions--turning equations into pictures--by building a table of sample points. Choose some points for one variable and compute the other variable. This will give the addresses of some of the points on the curve. If you have enough points you can sketch the function. The picture of the function can tell you a lot about the function. For example, the roots of the equation are the places where the curve crosses the X axis and if the roots are p, q and r the factors of the function are X - p, X - q and X - r. Also points where the curve changes directions indicate maximum and minimums of the function.

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