document.write( "Question 1027806: Please help me and show all steps and work. I am really lost and my professor really is no help. I have been trying to figure this out for days now. I never had any of these types of problems in highschool. Thanks\r
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\n" ); document.write( "\n" ); document.write( " Find the equation of the parabola that contains the points (-1,10),(2,-5)and(3,-18). Once you found the equation, answer the following questions:
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Algebra.Com's Answer #643016 by Edwin McCravy(20056) $\"\"$ $\"About$

You can put this solution on YOUR website!

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document.write( "The graph of any parabola has the form\r\n" );
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document.write( "y = f(x) = Ax² + Bx + C\r\n" );
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document.write( "(-1,10),(2,-5)and(3,-18). \r\n" );
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document.write( "Since it goes through (-1,10), substitute x=-1 and y=f(x)=10\r\n" );
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document.write( "f(x) = Ax² + Bx + C\r\n" );
document.write( "  10 = A(-1)² + B(-1) + C\r\n" );
document.write( "  10 = 1A - B + C\r\n" );
document.write( "A - B + C = 10\r\n" );
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document.write( "Since it goes through (2,-5), substitute x=2 and y=f(x)=-5\r\n" );
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document.write( "f(x) = Ax² + Bx + C\r\n" );
document.write( "  -5 = A(2)² + B(2) + C\r\n" );
document.write( "  -5 = 4A + 2B + C\r\n" );
document.write( "4A + 2B + C = -5\r\n" );
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document.write( "Since it goes through (3,-18), substitute x=3 and y=f(x)=-18\r\n" );
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document.write( "f(x) = Ax² + Bx + C\r\n" );
document.write( " -18 = A(3)² + B(3) + C\r\n" );
document.write( " -18 = 9A + 3B + C\r\n" );
document.write( "9A + 3B + C = -18\r\n" );
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document.write( "So we have this system of 3 equations and 3 unknowns:\r\n" );
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document.write( "Solve that system either by elimination or substitution\r\n" );
document.write( "and get A=-2, B=-3, C=9\r\n" );
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document.write( "So the equation  \r\n" );
document.write( "f(x) = Ax² + Bx + C\r\n" );
document.write( "becomes\r\n" );
document.write( "f(x) = -2x² - 3x + 9\r\n" );
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document.write( "The vertex is the point \r\n" );
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document.write( "So the vertex is \r\n" );
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document.write( "Since the coefficient of A is negative the parabola opens\r\n" );
document.write( "downward, and the vertex is a MAXIMUM. The focus is BELOW\r\n" );
document.write( "the vertex.\r\n" );
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document.write( "Since A=-2, the focus is 2 units BELOW the vertex, so we add\r\n" );
document.write( "A=-2 units to the y-coordinate of the vertex:\r\n" );
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document.write( "For the focus, we use the same x-coordinate as the vertex\r\n" );
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document.write( "So the vertex is  \r\n" );
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document.write( "The x-intercepts are found by setting y = 0:\r\n" );
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document.write( "         f(x) = -2x² - 3x + 9\r\n" );
document.write( "            0 = -2x² - 3x + 9\r\n" );
document.write( " 2x² + 3x - 9 = 0\r\n" );
document.write( "  (2x-3)(x+3) = 0\r\n" );
document.write( "  2x-3=0;  x+3=0\r\n" );
document.write( "    2x=3;    x=-3\r\n" );
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document.write( "So the x-intercepts are the points\r\n" );
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document.write( " and \r\n" );
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document.write( "Here is the graph.  The points marked are the vertex, focus,\r\n" );
document.write( "and x-intercepts.\r\n" );
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document.write( "Edwin

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