document.write( "Question 1194459: Find a quadratic model in standard form for (0,0), (1,-5), (2,0) \n" ); document.write( "

Algebra.Com's Answer #826662 by ikleyn(52787) $\"\"$ $\"About$

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\n" ); document.write( "Find a quadratic model in standard form for (0,0), (1,-5), (2,0)
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document.write( "You are looking to find a quiadratic finction in the form\r\n" );
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document.write( "    y = ax^2 + bx + c    (1)\r\n" );
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document.write( "such that its plot goes through three given points.\r\n" );
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document.write( "Notice that two given points, (0,0) and (2,0), have the same (identical) y-values.\r\n" );
document.write( "Moreover, these y-values are zeros, so these points are two y-intercepts of the parabola.\r\n" );
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document.write( "Hence, the parabola's symmetry axis is half-way between x-coordinates of these points:\r\n" );
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document.write( "     =  = 1.\r\n" );
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document.write( "So, the parabola in the vertex form is  \r\n" );
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document.write( "    y = ,     (2)\r\n" );
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document.write( "where  is the vertex' y-coordinate.\r\n" );
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document.write( "From the other side, you see that the second given point has x-coordinate equal to 1 - hence,\r\n" );
document.write( "this point is the vertex;  so you conclude that in formula (2)   = -5.\r\n" );
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document.write( "Thus you can write the parabola's expression in the vertex form\r\n" );
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document.write( "    y = .    (3)\r\n" );
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document.write( "Now, to determine the unknown coefficient \"a\", substitute x= 0 into formula (3) and use\r\n" );
document.write( "the given info y(0) = 0  (the coordinates of the first point)\r\n" );
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document.write( "    0 = ,\r\n" );
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document.write( "or\r\n" );
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document.write( "    0 = a - 5,\r\n" );
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document.write( "which gives you \r\n" );
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document.write( "    a = 5.\r\n" );
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document.write( "Now the parabola in vertex form is  y = .\r\n" );
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document.write( "To get the standard form, make FOIL\r\n" );
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document.write( "    y(x) = 5x^2 - 10x.\r\n" );
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document.write( "It is your ANSWER.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "It is one of possible ways to analyse.\r
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\n" ); document.write( "\n" ); document.write( "There is another way: it is to write the quadratic form in the form\r
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\n" ); document.write( "\n" ); document.write( " y = $\"ax%2A%28x-2%29\"$ ,\r
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\n" ); document.write( "\n" ); document.write( "based on the fact that x= 0 and x= 2 are the zeroes, and then to find the coefficient \" a \",
\n" ); document.write( "using info about the point (1,-5).\r
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