document.write( "Question 356772: find the vertex of the x and y coordinate, line of symmetry and maximum of f(x)
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Algebra.Com's Answer #254763 by Edwin McCravy(20055)\"\" \"About 
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find the vertex of the x and y coordinate, line of symmetry and maximum of f(x)
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document.write( "There are two methods, {1) completing the square using the memorized\r\n" );
document.write( "standard form f(x) = ax²+b+c \r\n" );
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document.write( "and \r\n" );
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document.write( "(2) using the vertex formula that you have memorized.\r\n" );
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document.write( "Method (1)\r\n" );
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document.write( "f(x) = -2x² + 2x + 7\r\n" );
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document.write( "Factor out -2 from the first two terms:\r\n" );
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document.write( "f(x) = -2(x² - x) + 7\r\n" );
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document.write( "To the side multiply the coefficient of x, which is -1, by \"1%2F2\" getting\r\n" );
document.write( "\"-1%2F2\" then squaring \"-1%2F2\" getting \"%28-1%2F2%29%5E2\" or \"1%2F4\".  Then\r\n" );
document.write( "adding that and subtracting that inside of the parentheses:\r\n" );
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document.write( "f(x) = -2(x² - x + \"1%2F4\" - \"1%2F4\") + 7\r\n" );
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document.write( "Change the parentheses to brackets (so you can put parentheses inside):\r\n" );
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document.write( "f(x) = -2[x² - x + \"1%2F4\" - \"1%2F4\"] + 7\r\n" );
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document.write( "Factor the first three terms inside the brackets:\r\n" );
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document.write( "f(x) = -2[(x - \"1%2F2\")(x - \"1%2F2\") - \"1%2F4\"] + 7\r\n" );
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document.write( "Since those factors in parentheses are the same we write them\r\n" );
document.write( "as a perfect square:\r\n" );
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document.write( "f(x) = -2[(x - \"1%2F2\")² - \"1%2F4\"] + 7\r\n" );
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document.write( "Now remove the brackets by using the distributive\r\n" );
document.write( "principle.  That is, we multiply the -2 by putting\r\n" );
document.write( "it in front of the (x - \"1%2F4\")^2 and we multiply\r\n" );
document.write( "the -2 also by the \"-1%2F2\" getting -1. So we have\r\n" );
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document.write( "f(x) = -2(x - \"1%2F2\")² + \"1%2F2\" + 7\r\n" );
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document.write( "Then we combine the two terms on the right side and get\r\n" );
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document.write( "f(x) = -2(x - \"1%2F2\")² + \"1%2F2\" + \"14%2F2\"\r\n" );
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document.write( "f(x) = -2(x - \"1%2F2\")² + \"15%2F2\"\r\n" );
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document.write( "We recognize this as in the standard form we have memorized:\r\n" );
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document.write( "f(x) = a(x - h)² + k\r\n" );
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document.write( "we know that a = -2, h = \"1%2F2\" and k = \"15%2F2\"\r\n" );
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document.write( "So that the vertex is (h,k) or (\"1%2F2\", \"15%2F2\") \r\n" );
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document.write( "Method (2), using the vertex formula we have memorized:\r\n" );
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document.write( "x-coordinate of vertex = \"-b%2F%282a%29\"\r\n" );
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document.write( "y-coordinate of vertex = what you get when you substitute the\r\n" );
document.write( "x-coordinate for x in the equation and simplify.\r\n" );
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document.write( "f(x) = -2x² + 2x + 7\r\n" );
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document.write( "Compare to the general form we have memorized,\r\n" );
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document.write( "f(x) = ax² + bx + c\r\n" );
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document.write( "a = -2, b = 2, c = 7\r\n" );
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document.write( "x-coordinate of vertex = \"-b%2F%282a%29\" = \"-%282%29%2F%282%28-2%29%29\" = \r\n" );
document.write( "\"%28-2%29%2F%28-4%29\" = \"1%2F2\"\r\n" );
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document.write( "y-coordinate of vertex = what we get when you substitute the\r\n" );
document.write( "x-coordinate, \"1%2F2\" for x in the equation and simplify:\r\n" );
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document.write( "f(\"1%2F2\") = -2(\"1%2F2\"² + 2\"1%2F2\" + 7 = -2(\"1%2F4\") + 1 + 7 = \"-1%2F2\" + 8 = \"-1%2F2\" + \"16%2F2\" = \"15%2F2\"\r\n" );
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document.write( "so the vertex is the point V(\"1%2F2\", \"15%2F2\").\r\n" );
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document.write( "No that we have found the vertex by either of the above two methods,\r\n" );
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document.write( "we plot that point and draw a verticle line, the line of symmetry,\r\n" );
document.write( "through it, like this line drawn in green:\r\n" );
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document.write( "That green line of symmetry has the equation x = \"1%2F2\" because\r\n" );
document.write( "every point on that green line of symmetry has \"1%2F2\" as its \r\n" );
document.write( "x-coordinate.\r\n" );
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document.write( "Now we can get some other points on that graph:\r\n" );
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document.write( " x| y\r\n" );
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document.write( "-2|-5 \r\n" );
document.write( "-1| 3\r\n" );
document.write( " 0| 7\r\n" );
document.write( " 1| 7\r\n" );
document.write( " 2| 3\r\n" );
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document.write( "And the graph is\r\n" );
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document.write( "The maximum value is the greatest y-value on the graph, which is the\r\n" );
document.write( "y-coordinate of the vertex, \"15%2F2\"\r\n" );
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document.write( "Edwin
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