document.write( "Question 558739: What is the vertex, focus, axis of symmetry, and directiv of the following equation, (x-2)² = y+3 and how did you get those answers? \n" ); document.write( "
Algebra.Com's Answer #363168 by Edwin McCravy(20055) $\"\"$ $\"About$
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document.write( "The standard form of a parabola whose axis is symmetry is vertical is\r\n" );
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document.write( "(x - h)² = 4p(y - k)      [Some books use \"a\" or \"c\" instead of \"p\"]\r\n" );
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document.write( "Where (h,k) is the vertex.  |p| is the diatance from the vertex to\r\n" );
document.write( "the focus (which is a point inside the parabola on its axis of symmetry),\r\n" );
document.write( "and also to the directrix, which is a line outside the parabola \r\n" );
document.write( "perpendicular to its line of symmetry.  If p is positive the parabola\r\n" );
document.write( "opens upward, and if p is negative it opens downward.\r\n" );
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document.write( "We compare your equation to that one:\r\n" );
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document.write( "(x - 2)² = y + 3\r\n" );
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document.write( "To get it looking like \r\n" );
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document.write( "(x - h)² = 4p(y - k)\r\n" );
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document.write( "we put parentheses around the right side and a 1 infront\r\n" );
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document.write( "(x - 2)² = 1(y + 3)\r\n" );
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document.write( "So we see that h = 2, k = -3, and 4p = 1 which makes p = \r\n" );
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document.write( "So the vertex is (h,k) = (2,-3).\r\n" );
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document.write( "We plot the vertex (2,-3), and draw a green axis of symmetry through\r\n" );
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document.write( "That green axis of symmetry goes through x = 2, so that's its equation.\r\n" );
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document.write( "The vertex is a point p or  of a unit above the vertex. It is \r\n" );
document.write( "on the axis of symmetry so it's x-coordinate is the same as the x-coordinate\r\n" );
document.write( "of the vertex, which is 2, but its y-coordinate is  of a unit\r\n" );
document.write( "more, so we add  to the y-coordinate of the vertex:\r\n" );
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document.write( "-3+ = + = , \r\n" );
document.write( "So the focus has the coordinates (2,)\r\n" );
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document.write( "We draw the focus:\r\n" );
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document.write( "The directrix is a horizontal line p or  of a unit below the vertex    \r\n" );
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document.write( "We draw it in blue:\r\n" );
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document.write( "Since the line is  unit below the vertex, we subtract  from\r\n" );
document.write( "its y-coordinate -3- = - = ,\r\n" );
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document.write( "so the equation of the directrix is y = \r\n" );
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document.write( "We draw two adjacent squares, with a common side from the directrix\r\n" );
document.write( "to the focus, like this:\r\n" );
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document.write( "and sketch in the parabola through the upper corners of those squares and\r\n" );
document.write( "through the vertex:\r\n" );
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document.write( "Edwin
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