\r\n" );
document.write( "The graph of a circle describes y as a function of x. \r\n" );
document.write( "True \r\n" );
document.write( "False \r\n" );
document.write( "\r\n" );
document.write( "This is false because in order to have a function, it\r\n" );
document.write( "must not be possible to draw a vertical line and have\r\n" );
document.write( "it touch or cross the graph more than just ONE point.\r\n" );
document.write( "\r\n" );
document.write( "Take this graph of a circle:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now let's arbitrarily draw a blue vertical line down \r\n" );
document.write( "through the graph:\r\n" );
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document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Notice that it intersects the circle at two points one \r\n" );
document.write( "above the x-axis and one below the x-axis. This is not\r\n" );
document.write( "allowed for the graph of a function, so a circle does not\r\n" );
document.write( "represent a function, i.e., does not describe y as a\r\n" );
document.write( "function.\r\n" );
document.write( "\r\n" );
document.write( "-------------------\r\n" );
document.write( "\r\n" );
document.write( "To answer the next questions, you must memorize that\r\n" );
document.write( "the standard equation of the circle with center (h,k) \r\n" );
document.write( "and radius r is\r\n" );
document.write( "\r\n" );
document.write( "(x - h)² + (y - k)² = r²\r\n" );
document.write( "\r\n" );
document.write( "-------------------\r\n" );
document.write( "\r\n" );
document.write( "Which of these equations describes a circle\r\n" );
document.write( "with radius 5 and center (3,-2) ?\r\n" );
document.write( "(x - 3)² + (y + 2)² = 25 \r\n" );
document.write( "(x - 3)² + (y + 2)² = 5 \r\n" );
document.write( "(x + 3)² + (y - 2)² = 25 \r\n" );
document.write( "(x + 3)² - (y + 2)² = 25\r\n" );
document.write( "\r\n" );
document.write( "Substitute h=3, k=-2, r=5 into\r\n" );
document.write( "\r\n" );
document.write( "(x - h)² + (y - k)² = r²\r\n" );
document.write( "(x - 3)² + (y -(-2) )² = 5²\r\n" );
document.write( "(x - 3)² + (y + 2)² = 25\r\n" );
document.write( "So it's the first choics.\r\n" );
document.write( "\r\n" );
document.write( "-----------------------\r\n" );
document.write( "\r\n" );
document.write( "Which of these equations describes a circle\r\n" );
document.write( "with radius 3/2 and center (0,4) ?\r\n" );
document.write( "x² - (y - 4)² = 9/4 \r\n" );
document.write( "x² + (y - 4)² = 3/2 \r\n" );
document.write( "(x - 4)² + y² = 9/4 \r\n" );
document.write( "x² + (y - 4)² = 9/4\r\n" );
document.write( "\r\n" );
document.write( "Substitute h=0 k=4 r=3/2 into\r\n" );
document.write( "\r\n" );
document.write( "(x - h)² + (y - k)² = r²\r\n" );
document.write( "(x - 0)² + (y - 4)² = (3/2)²\r\n" );
document.write( " x² + (y - 4)² = 9/5\r\n" );
document.write( "So it's the fourth choice.\r\n" );
document.write( "\r\n" );
document.write( "------------------------\r\n" );
document.write( "\r\n" );
document.write( "The circle whose equation is \r\n" );
document.write( "x² + 2x + y² + 6y + 6 = 0\r\n" );
document.write( "\r\n" );
document.write( "has center (1,3) and radius 2 \r\n" );
document.write( "has center (-1, -3) and radius 2 \r\n" );
document.write( "has center (-1, -3) and radius 4 \r\n" );
document.write( "has center (1, 3) and radius 4\r\n" );
document.write( "\r\n" );
document.write( "Now you must learn to change from standard form\r\n" );
document.write( "of a circle (where there are no parentheses) to \r\n" );
document.write( "the standard form (x - h)² + (y - k)² = r². We\r\n" );
document.write( "do this by completing the square, very similar \r\n" );
document.write( "to the way you completed the square back when\r\n" );
document.write( "you were solving quadratic equations:\r\n" );
document.write( "\r\n" );
document.write( " x² + 2x + y² + 6y + 6 = 0\r\n" );
document.write( "\r\n" );
document.write( "We get the constant term, 6, off the left side by\r\n" );
document.write( "subtrracting 6 from both sides:\r\n" );
document.write( "\r\n" );
document.write( "x² + 2x + 1 + y² + 6y + 9 = -6 + 1 + 9\r\n" );
document.write( "\r\n" );
document.write( "The x terms and the y terms are together, so we\r\n" );
document.write( "skip a space after the 2x and after the 6y\r\n" );
document.write( "\r\n" );
document.write( "x² + 2x + y² + 6y = -6 \r\n" );
document.write( "\r\n" );
document.write( "Now we multiply the coefficient of x, which is 2, by 1/2\r\n" );
document.write( "That gives us 1.\r\n" );
document.write( "Now we square that result.\r\n" );
document.write( "1² = 1. So we add +1 to both sides of the equation. On\r\n" );
document.write( "the left we put it in the first space we skipped. We add\r\n" );
document.write( "it on the end of the right side;\r\n" );
document.write( "\r\n" );
document.write( "x² + 2x + 1 + y² + 6y = -6 + 1 \r\n" );
document.write( "\r\n" );
document.write( "Now we multiply the coefficient of y which is 6 by 1/2\r\n" );
document.write( "That gives us 3.\r\n" );
document.write( "Now we square that result.\r\n" );
document.write( "3² = 9 So we add +9 to both sides of the equation. On\r\n" );
document.write( "the left we put it in the second space we skipped. We add\r\n" );
document.write( "it on the end of the right side;\r\n" );
document.write( "\r\n" );
document.write( "x² + 2x + 1 + y² + 6y + 9 = -6 + 1 + 9\r\n" );
document.write( "\r\n" );
document.write( "Now we factor the first three terms x² + 2x + 1 = \r\n" );
document.write( "(x + 1)(x + 1) = (x + 1)²\r\n" );
document.write( "\r\n" );
document.write( "Now we factor the last three terms y² + 6x + 9 = \r\n" );
document.write( "(y + 3)(y + 3) = (y + 3)²\r\n" );
document.write( "\r\n" );
document.write( "Now we combine the numbers on the right\r\n" );
document.write( " -6 + 1 + 9 = 4\r\n" );
document.write( "\r\n" );
document.write( "So we now have\r\n" );
document.write( "\r\n" );
document.write( "(x + 1)² + (y + 3)² = 4\r\n" );
document.write( "\r\n" );
document.write( "Now we compare this with the standard\r\n" );
document.write( "form:\r\n" );
document.write( "\r\n" );
document.write( "(x - h)² + (y - k)² = r²\r\n" );
document.write( "\r\n" );
document.write( "We see that \"+ 1\" is in the place of \"- h\", so\r\n" );
document.write( "- h = + 1 and therefore h = - 1\r\n" );
document.write( "\r\n" );
document.write( "We see that \"+ 3\" is in the place of \"- k\", so\r\n" );
document.write( "- k = + 3 and therefore h = - 3\r\n" );
document.write( "\r\n" );
document.write( "So the center is (h, k) = (-1, -3)\r\n" );
document.write( "\r\n" );
document.write( "We see that \"4\" is in the place of \"r²\", so\r\n" );
document.write( "r² = 4, and therefore r = 2, or its radius is 2\r\n" );
document.write( "\r\n" );
document.write( "So the answer is the third choice\r\n" );
document.write( "\"has center (-1, -3) and radius 2\" \r\n" );
document.write( "\r\n" );
document.write( "-------------------\r\n" );
document.write( "\r\n" );
document.write( "The circle whose equation is x² - 6x + y² + 2y = -4\r\n" );
document.write( "\r\n" );
document.write( "You forgot to list the choices but you can find out \r\n" );
document.write( "what its center and radius are. Follow the steps \r\n" );
document.write( "above and you'll get\r\n" );
document.write( "\r\n" );
document.write( "(x - 3)² + (y + 1)² = 6\r\n" );
document.write( " _\r\n" );
document.write( "and its center is (3,-1) and its radius is Ö6\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
