Lecture-12

Lecture-12 - Lecture 12: Changing coordinates and...

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Unformatted text preview: Lecture 12: Changing coordinates and transforming graphs June 17, 2010 Introduction Suppose ( x,y ) is a statement in which the variables x and y appear. For example, ( x,y ) might be the statement x 2 + y 2 = 35 2 . It could also be a more complicated statement, such as: x 2 + y 2 = 35 2 and y = 2 3 ( x + 1) , or x 2 + y 2 = 35 2 and x and y are both rational . No matter how complicated ( x,y ) is, by the graph of ( x,y ), we mean the set of all number pairs for which ( x,y ) is true. If we interpret number pairs as points by referring to the standard x- y-coordinate system on the plane, then the graph of ( x,y ) is a set of points in the plane. 1 We will use the symbol X to denote the graph: X := { ( x,y ) | ( x,y ) } . Now, suppose a and b are any numbers. Consider the set X := { ( x,y ) | ( x- a,y- b ) } . I assert that X coincides with the set of all pairs obtained by taking a pair belonging to X and adding ( a,b ) to it. In other words, X is the image of X after a translation by ( a,b ). We can demonstrate this assertion as follows: ( ( x + a ) , ( y + b ) ) X ( ( x + a )- a, ( y + b )- b ) ( x ,y ) ( x ,y ) X Returning to the example of the circle,...
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This note was uploaded on 11/23/2011 for the course MATH 6302 taught by Professor Madden during the Summer '11 term at LSU.

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Lecture-12 - Lecture 12: Changing coordinates and...

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