71Notes - 1 Math 71 Outline Lecture Notes (Prepared by...

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1 Math 71 Outline Lecture Notes (Prepared by Stefan Waner) 1. The Cartesian Plane and Distance The coordinate plane is the infinite plane with two perpendicular axes—the x -axis and the y -axis. The axes divide the plane into four quadrants—shown in class. To locate a point, we use coordinates, so that each point is represented by a pair (a, b) of real numbers. Example A. Locate the points P(2,3), Q(-4,3), R(-5,-2), S(4,-3), T(0,-3) and U(4,0) B. Sketch the following regions: x = 4; |x| 5, x+y 1. Distance Between Two Points To find the distance between the general points P 1 (x 1 , y 1 ) and P 2 ( x 2 , y 2 ), introduce the third point Q (x 2 , y 1 ) as shown, and use Pythagoras' Theorem (shown in class). We obtain the formula: d 2 = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 giving d = (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 Distance Formula Examples A. Find the distance between P(1,2) and Q(-1,6) . B. Find the distance from the general point (a, b) to the origin—gives another very important formula. 2. Equations and Graphs An equation is a mathematical expression containing an “equals” sign and satisfying various syntactical rules. Eg. x+y = 8 ; x = 5, 1 = 0. Given an equation in x and y , we can represent it by a “curve” in the coordinate plane. Formally: The graph of an equation consists of all points (x, y) whose coordinates satisfy the equation. To say that the coordinates of a point “satisfy” the equation means that when you substitute them as values for x and y, the equation expresses a truth about numbers. Examples A. Graph the equation x+y = 6. B. Graph the equation y = x 2 . C. Graph the equation x 2 +y 2 = 9. D. Obtain the equation of a general circle center (a, b) radius r as: (x - a) 2 + (y - b) 2 = r 2 Circle Center (a, b) Radius r E. Sketch the graph of the equation x 2 + y 2 + 2x - 6y + 7 = 0 . F. Find the equation of the upper semicircle with radius 2 centered at (3, -1). 3. Straight Lines The slope of a straight line is a measure of its steepness, and is defined as the number of units it goes up for every unit across (in the +x direction). In terms of a formula, m = y 2 - y 1 x 2 - x 1 = y x Slope Formula given two points (x 1 , y 1 ) and (x 2 , y 2 ) on that line. Example Find the slope of the line through (1,2) and (2,1).
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2 Facts we check in class A. Check that it is independent of the choice of points used in calculating it. B. Parallel lines have the same slope C. Perpendicular lines have negative reciprocal slopes. D. For the point (x, y) to lie on the line through (x 0 , y 0 ) and have slope m, it must satisfy the equation (y - y 0 ) = m(x - x 0 ) Equation of a Line—Point-Slope Formula Thus to get the equation of a line, we need only two things: (1) a point on it; (2) its slope. E. Slope = tan ø . Examples A. Line through (0,1) with slope 6. B. Line through (1,2) and (2,1) C. Line with slope m and y-intercept b gives the “slope-intercept” formula: y = mx + b Equation of a Line—Slope-Intercept Formula D. Line through ( 4,-1) parallel to y = -2x + 4.
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This note was uploaded on 04/29/2008 for the course MATH 71 taught by Professor Waner during the Spring '08 term at Hofstra University.

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71Notes - 1 Math 71 Outline Lecture Notes (Prepared by...

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