document.write( "Question 89451: Please help me on how to figure out the vertex of the equation 2y=x2(this is x squared)-2x-5. \n" ); document.write( "
Algebra.Com's Answer #65033 by Edwin McCravy(20086)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "2y = x² - 2x - 5\r\n" );
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document.write( "Method 1:\r\n" );
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document.write( "Get it in the form\r\n" );
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document.write( "    y = a(x - h)² + k\r\n" );
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document.write( "where the vertex is (h,k) and the two neighbor\r\n" );
document.write( "points are (h±1,k+a)\r\n" );
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document.write( "Take one-half of the coefficient of x, which is \r\n" );
document.write( "one-half of -2 or -1.  Then square -1 and get\r\n" );
document.write( "+1 then add +1  to both sides\r\n" );
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document.write( "2y + 1 = x² - 2x + 1 - 5\r\n" );
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document.write( "Factor the first three terms on the right\r\n" );
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document.write( "2y + 1 = (x - 1)(x - 1) - 5\r\n" );
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document.write( "Write (x - 1)(x - 1) as (x - 1)²\r\n" );
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document.write( "2y + 1 = (x - 1)² - 5\r\n" );
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document.write( "Add -1 to both sides:\r\n" );
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document.write( "    2y = (x - 1)x² - 6  \r\n" );
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document.write( "Divide every term by 2:\r\n" );
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document.write( "    y = \"1%2F2\"(x - 1)² - 3\r\n" );
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document.write( "Compare to the standard equation:\r\n" );
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document.write( "    y = a(x - h)² + k\r\n" );
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document.write( "and we see that a = \"1%2F2\", h = 1, k= -3\r\n" );
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document.write( "So the vertex is\r\n" );
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document.write( "(h,k) = (1,-3) \r\n" );
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document.write( "That's all you wanted but you should take\r\n" );
document.write( "this opportunity to learn to graph.\r\n" );
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document.write( "The two neighbor points\r\n" );
document.write( "are (h±1,k+a) = (1±1, -3+\"1%2F2\") which are\r\n" );
document.write( "(0,\"-5%2F2\") and (2,\"-5%2F2\")\r\n" );
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document.write( "We can draw the graph by plotting those three \r\n" );
document.write( "points first\r\n" );
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document.write( "Then sketch a parabola through those three \r\n" );
document.write( "points:\r\n" );
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document.write( "Another way just to find the vertex is to\r\n" );
document.write( "memorize the vertex formula:\r\n" );
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document.write( "----------------------\r\n" );
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document.write( "The general equation y = ax² + bx + c\r\n" );
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document.write( "has vertex (h,k) where\r\n" );
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document.write( "h = \"-b%2F%282a%29\"\r\n" );
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document.write( "k = the value of y when h is substituted for x\r\n" );
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document.write( "-----------------------\r\n" );
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document.write( "So for \r\n" );
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document.write( "2y = x² - 2x - 5\r\n" );
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document.write( "Get it in general form  y = ax² + bx + c\r\n" );
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document.write( "Divide through every term by 2\r\n" );
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document.write( "y = \"%281%2F2%29x%5E2+-+x+-+5%2F2\"\r\n" );
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document.write( "Compare to the general form\r\n" );
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document.write( "y = ax² + bx + c\r\n" );
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document.write( "and we see that\r\n" );
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document.write( "a = \"1%2F2\", b = -1, c = \"-5%2F2\"\r\n" );
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document.write( "h = \"-b%2F%282a%29\" = \"-%28%28%28-1%29%29%2F%282%281%2F2%29%29%29\" = \"-%28%28%28-1%29%29%2F1%29%29\" = \"-%28-1%29\" = 1\r\n" );
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document.write( "k = the value of y when 1 is substituted for x \r\n" );
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document.write( "y = \"%281%2F2%29x%5E2+-+x+-+5%2F2\"\r\n" );
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document.write( "y = \"%281%2F2%291%5E2+-+1+-+5%2F2\" = \"1%2F2-1-5%2F2\" = \"-4%2F2-1\" = \"-2-1\" = \"-3\"\r\n" );
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document.write( "So k = -3\r\n" );
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document.write( "So the vertex by this method is also (h,k) = (1,-3)\r\n" );
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document.write( "Edwin

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