document.write( "Question 628327: how do you graph x<6y-y^2 \n" ); document.write( "
Algebra.Com's Answer #395564 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "x < 6y - y²\r\n" );
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document.write( "Graph the boundary curve by replacing the < by =\r\n" );
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document.write( "x = 6y - y²\r\n" );
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document.write( "y² - 6y = -x\r\n" );
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document.write( "Complete the square on the left by adding (·(-3)² = 9\r\n" );
document.write( "to both sides:\r\n" );
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document.write( "y² - 6y + 9 < -x + 9\r\n" );
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document.write( "Factor the left side\r\n" );
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document.write( "(y - 3)² = -x + 9\r\n" );
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document.write( "This is in the form of a parabola with\r\n" );
document.write( "horizontal line of symmetry\r\n" );
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document.write( "(y - k)² = 4p(x - h)\r\n" );
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document.write( "(y - 3)² = -1(x - 9)\r\n" );
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document.write( "The vertex is (h,k) = (9,3) and it opens left since 4p = -1\r\n" );
document.write( "p =  which is negative:\r\n" );
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document.write( "You can get a few points in addition to the vertex, say\r\n" );
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document.write( "(5,5), (5,1), (0,6), (0,0), (-7,7), (-7,-1)\r\n" );
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document.write( "  \r\n" );
document.write( "Now the question is whether to shade the inside portion or the outside\r\n" );
document.write( "portion of the parabola.  So we test a point that's not on the parabola,\r\n" );
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document.write( "say (3,4), which is inside the parabola, in the original inequality:\r\n" );
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document.write( "We substitute that into the original inequality:\r\n" );
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document.write( "x < 6y - y²\r\n" );
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document.write( "3 < 6(4) - (4)²\r\n" );
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document.write( "3 < 24 - 16\r\n" );
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document.write( "3 < 8\r\n" );
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document.write( "This is true, so since (3,4) is a solution and it is on the\r\n" );
document.write( "inside of the parabola, all the points inside the parabola\r\n" );
document.write( "are solutions.\r\n" );
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document.write( "Edwin
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