Geometric Objects and Properties using AlgebraDate: 06/29/98 at 00:27:27 From: Suzie Liu Subject: Algebra 1 My teacher asked us why y = mx+b, and what does it mean. Date: 06/29/98 at 14:43:14 From: Doctor Rob Subject: Re: Algebra 1 Look at a plane. Pick a point O called the origin. Through O draw a line called the x-axis (usually horizontal). At O construct a perpendicular line called the y-axis. Mark a point on the x-axis, labeled "1" (usually to the right of O). The distance from O to 1 will be one unit. The direction moving from O to 1 will be the positive direction along the x-axis, and the opposite direction will be the negative one. The points on the x-axis now correspond to real numbers and vice versa. In the same way, choose a direction (usually up) and a unit distance (often the same as on the x-axis) on the y-axis. Points on the y-axis also correspond to real numbers and vice versa. This setup is called the Cartesian plane. y-axis ^ | | 1- | | ---------------o------|-------> x-axis O| 1 | | | | Now given any point P in the plane, drop a perpendicular to the x-axis, meeting it at Q, and a perpendicular to the y-axis, meeting it at R. Then the real number x corresponding to Q is called the x-coordinate (or abscissa) of P, and the real number y corresponding to R is called the y-coordinate (or ordinate) of P. Thus the point P corresponds to a pair of numbers (x,y). Likewise, a pair of numbers (x,y) corresponds to a point P, since given x and y, you can find the point Q corresponding to the number x on the x-axis, and the point R corresponding to the number y on the y-axis. Now erect a perpendicular to the x-axis at Q and a perpendicular to the y-axis at R. They will intersect in a unique point P. This is called a Cartesian coordinate system. Points correspond to pairs of real numbers called the coordinates of the point. y-axis ^ | | 1- | | x Q --------------------o------|-------o----> x-axis O | 1 | |y |y | | | | R o--------------o P | x Now a line will consist of a set of points, that is, a set of pairs (x,y). Which set is determined by an equation. All pairs of real numbers (x,y) that satisfy the equation y = m*x + b, where m and b are fixed, given real numbers, are the coordinates of points lying on a line not parallel to the y-axis. The number b is the real number corresponding to the point on the y-axis where the line crosses it, called the y-intecept. That point has coordinates (0,b). These values of x and y do satisfy the above equation, as you can check for yourself. The number m is called the slope of the line, and represents the amount the y-coordinate changes when we increase the x-coordinate by exactly 1. For example, the point (1,b+m) lies on the line, because these values satisfy the equation (again, check this for yourself), and the y-coordinate has increased by m while the x-coordinate has increased by 1 as we move from (0,b) to (1,b+m) along the line. If m is positive the line rises as we move from left to right. The bigger m is, the steeper the rise of the line is. If m is negative, the line falls as we move from left to right. The bigger the absolute value of m, the steeper the decline of the line is. Lines are parallel to the y-axis are vertical. They have no slope, and their equation is of the form x = a, for some fixed, given real number a. This is a basic introduction to a subject called Analytical Geometry. It was invented by Rene' Descartes, a French mathematician of the 17th century. It amounts to a way to talk about geometric objects and properties using algebra. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/ |
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