\r\n" );
document.write( "4x² - 8x - y² + 4y - 4 = 0\r\n" );
document.write( "\r\n" );
document.write( "Factor the coefficient of x², which is 4, out of \r\n" );
document.write( "the two x-terms \r\n" );
document.write( "\r\n" );
document.write( "4(x² - 2x) - y² + 4y - 4 = 0\r\n" );
document.write( "\r\n" );
document.write( "Factor the coefficient of y², which is -1, out of \r\n" );
document.write( "the two y-terms\r\n" );
document.write( "\r\n" );
document.write( "4(x² - 2x) - (y² - 4y) - 4 = 0\r\n" );
document.write( "\r\n" );
document.write( "Get the term -4 off the left side by adding\r\n" );
document.write( "4 to both sides:\r\n" );
document.write( " \r\n" );
document.write( " 4(x² - 2x) - (y² - 4y) = 4\r\n" );
document.write( "\r\n" );
document.write( "Multiply the coefficient of x inside the parentheses,\r\n" );
document.write( "which is -2 by ½ getting -1. Then square -1 getting +1.\r\n" );
document.write( "Add +1 inside the first parentheses, and offset it by\r\n" );
document.write( "adding +4 to the right side, since adding +1 inside\r\n" );
document.write( "the first parentheses amounts to adding +4 to the left side\r\n" );
document.write( "because of the 4 outside the first parentheses:\r\n" );
document.write( "\r\n" );
document.write( "4(x² - 2x + 1) - (y² - 4y) = 4 + 4\r\n" );
document.write( "\r\n" );
document.write( "Combine the 4 + 4 on the right as 8\r\n" );
document.write( "\r\n" );
document.write( "4(x² - 2x + 1) - (y² - 4y) = 8\r\n" );
document.write( "\r\n" );
document.write( "Multiply the coefficient of y inside the second parentheses,\r\n" );
document.write( "which is -4 by ½ getting -2. Then square -2 getting +4.\r\n" );
document.write( "Add +4 inside the second parentheses, and offset it by\r\n" );
document.write( "adding -4 to the right side, since adding +4 inside\r\n" );
document.write( "the second parentheses amounts to adding -4 to the left side\r\n" );
document.write( "because of the - outside the second parentheses:\r\n" );
document.write( "\r\n" );
document.write( "4(x² - 2x + 1) - (y² - 4y + 4) = 8 - 4\r\n" );
document.write( "\r\n" );
document.write( "Combine the 8 - 4 on the right as 4\r\n" );
document.write( "\r\n" );
document.write( "4(x² - 2x + 1) - (y² - 4y + 4) = 4\r\n" );
document.write( "\r\n" );
document.write( "Factor x² - 2x + 1 as (x - 1)(x - 1) and then as (x - 1)²\r\n" );
document.write( "Factor y² - 4y + 4 as (y - 2)(y - 2) and then as (y - 2)²\r\n" );
document.write( "\r\n" );
document.write( " 4(x - 1)² - (y - 2)² = 4\r\n" );
document.write( "\r\n" );
document.write( "Next we must get a 1 on the right.\r\n" );
document.write( "So we divide all the terms by 4\r\n" );
document.write( "\r\n" );
document.write( " 4(x - 1)² (y - 2)² 4\r\n" );
document.write( " ————————— - ———————— = ——— \r\n" );
document.write( " 4 4 4\r\n" );
document.write( "\r\n" );
document.write( " (x - 1)² (y - 2)² \r\n" );
document.write( " ————————— - ———————— = 1 \r\n" );
document.write( " 1 4 \r\n" );
document.write( "\r\n" );
document.write( "Next we compare that to\r\n" );
document.write( "\r\n" );
document.write( " (x - h)² (y - k)² \r\n" );
document.write( " ————————— - ———————— = 1 \r\n" );
document.write( " a² b²\r\n" );
document.write( "\r\n" );
document.write( "which means that it is a hyperbolka that looks like this )(\r\n" );
document.write( "\r\n" );
document.write( "We see that h = 1, k = 2, a² = 1 so a = 1 and b² = 4 so b = 2\r\n" );
document.write( "\r\n" );
document.write( "The center = (h,k) = (1,2)\r\n" );
document.write( "\r\n" );
document.write( "Plot the center (1,2)\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Draw the transverse axis which is 2a or 2 units long and\r\n" );
document.write( "is horizontal, with the center at its midpoint. That \r\n" );
document.write( "means we draw 1 units left of the center and 1 units\r\n" );
document.write( "right of the center to get the complete transverse\r\n" );
document.write( "axis, drawn in green below:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Next draw the conjugate axis which is 2b or 4 units long and\r\n" );
document.write( "is vertical, with the center at its midpoint. That \r\n" );
document.write( "means we draw 2 units up from the center and 2 units\r\n" );
document.write( "down from the center to get the complete transverse\r\n" );
document.write( "axis, also drawn in green below:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Next we draw the defining rectangle, which is a\r\n" );
document.write( "rectangle with the ends of the transverse and\r\n" );
document.write( "conjugate axes as midpoints of the sides. I'll\r\n" );
document.write( "draw it in green, too:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( " \r\n" );
document.write( "Next we draw the extended diagonals of the defining rectangle.\r\n" );
document.write( "They will be the asymptotes of the hyperbola:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now we can sketch in the hyperbola to have its vertices as\r\n" );
document.write( "the ends of the transverse axis and to approach the \\r\n" );
document.write( "asymptotes. I'll draw the hyperbola in red:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "vertices, co-vertices, foci, and asymptotes\r\n" );
document.write( "\r\n" );
document.write( "The vertices are the endpoints of the transverse axis.\r\n" );
document.write( "They are (0,2) and (2,2)\r\n" );
document.write( "\r\n" );
document.write( "The co-vertices are the endpoints of the conjugate axis.\r\n" );
document.write( "They are (1,0) and (1,4)\r\n" );
document.write( "\r\n" );
document.write( "To find the foci we calculate c from the formula\r\n" );
document.write( "\r\n" );
document.write( "c² = a² + b²\r\n" );
document.write( "c² = 1² + 2²\r\n" );
document.write( "c² = 1 + 4\r\n" );
document.write( "c² = 5_\r\n" );
document.write( " c = √5\r\n" );
document.write( "\r\n" );
document.write( "Then the foci are the points which are c units right\r\n" );
document.write( "and left of the vertices. They are\r\n" );
document.write( " _ _ \r\n" );
document.write( "(1+√5,2) and (1-√5,2)\r\n" );
document.write( "\r\n" );
document.write( "I'll plot the two foci:\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "Now all we need are the equations of the two asymptotes:\r\n" );
document.write( "\r\n" );
document.write( "The asymptote that slants uphill to the right passes through (0,0)\r\n" );
document.write( "and (1,2),\r\n" );
document.write( "\r\n" );
document.write( "We use the slope formula 
\r\n" );
document.write( "\r\n" );
document.write( "Substitute in 
\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "y = 2x\r\n" );
document.write( "\r\n" );
document.write( "That's the equation of the asymptote that slants \r\n" );
document.write( "uphill to the right.\r\n" );
document.write( "\r\n" );
document.write( "The asymptote that slants downhill to the right\r\n" );
document.write( "passes through (2,0) and (1,2),\r\n" );
document.write( "\r\n" );
document.write( "Using the slope formula 
\r\n" );
document.write( "\r\n" );
document.write( "Substitute in \r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "y = -2x + 2\r\n" );
document.write( "\r\n" );
document.write( "That's the equation of the asymptote that slants\r\n" );
document.write( "downhill to the right.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" );
document.write( "![\"\"](\"/images/stars/stars-13.gif\")
