document.write( "Question 62168This question is from textbook Intermediate Algebra
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Algebra.Com's Answer #42927 by Edwin McCravy(20056) $\"\"$ $\"About$
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Find the vertex, axis of symmetry, maximum value or minimum \r\n" );
document.write( "value, x and y intercepts. \r\n" );
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document.write( "f(x) = -3/11(x - 7)² - 9\r\n" );
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document.write( "You must learn these facts about an equation of the form:\r\n" );
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document.write( "f(x) = a(x - h)² + k\r\n" );
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document.write( "1. The graph is shaped like this È if \" a \" is positive.\r\n" );
document.write( "   The graph is shaped like this Ç if \" a \" is negative.\r\n" );
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document.write( "2. The vertex is the lowest point on the graph if it looks \r\n" );
document.write( "   like this È \r\n" );
document.write( "   The vertex is the highest point on the graph if it looks \r\n" );
document.write( "   like this Ç\r\n" );
document.write( "   The vertex is the point (h, k)\r\n" );
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document.write( "3. The graph always goes through the points\r\n" );
document.write( "   (h + 1, k + a) and (h - 1, k + a)\r\n" );
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document.write( "4. The axis of symmetry is the vertical line which bisects\r\n" );
document.write( "   the parabola.  Its equation is x = h.\r\n" );
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document.write( "5. To find the y intercept, substitute x = 0 and solve for\r\n" );
document.write( "   y. Then the y-intercept is the point (0, that value of y)\r\n" );
document.write( "   There will always be a y-intercept.\r\n" );
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document.write( "6. 5. To find the x intercept(s), substitute y = 0 and solve for\r\n" );
document.write( "   x. Then the x-intercept(s) are the point \r\n" );
document.write( "   (that or those values of x, 0)\r\n" );
document.write( "   Sometimes there are no x-intercept(s).  This will be the case\r\n" );
document.write( "   if the value of x is imaginary. \r\n" );
document.write( "   \r\n" );
document.write( "Now using your equation:\r\n" );
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document.write( "f(x) = -3/11(x - 7)² - 9\r\n" );
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document.write( "compare that to \r\n" );
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document.write( "f(x) = a(x - h)² + k\r\n" );
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document.write( "a = -3/11, h = 7, k = -9\r\n" );
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document.write( "1. The graph is shaped like this Ç because\r\n" );
document.write( "   \" a \" is negative.\r\n" );
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document.write( "2. The vertex is the highest point on the graph. \r\n" );
document.write( "   The vertex is the point (h, k) = (7, -9)\r\n" );
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document.write( "3. The graph goes through the points\r\n" );
document.write( "   (h - 1, k + a) and (h + 1, k + a), which are\r\n" );
document.write( "   (7 - 1, -9 - 3/11) and (7 + 1, -9 - 3/11) or\r\n" );
document.write( "   (6, -102/11) and (8, -102/11) or about\r\n" );
document.write( "   (6, -9.3) and (8, -9.3)   \r\n" );
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document.write( "4. The axis of symmetry is the vertical line which bisects\r\n" );
document.write( "   the parabola.  Its equation is x = h or x = 7.\r\n" );
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document.write( "5. To find the y-intercept, we substitute x = 0 and solve for\r\n" );
document.write( "   f(0).\r\n" );
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document.write( "   f(x) = -3/11(x - 7)² - 9\r\n" );
document.write( "   f(0) = -3/11(0 - 7)² - 9\r\n" );
document.write( "   f(0) = -3/11(-7)² - 9\r\n" );
document.write( "   f(0) = -3/11(49) - 9\r\n" );
document.write( "   f(0) = -147/11 - 9\r\n" );
document.write( "   f(0) = -246/11\r\n" );
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document.write( "   So the y-intercept (0, -246/11), which\r\n" );
document.write( "   is about (0, -22.4) \r\n" );
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document.write( "6. 5. To find the x intercept(s), we substitute f(x) = 0 \r\n" );
document.write( "   and solve for x. Then the x-intercept(s) are the point \r\n" );
document.write( "   (that or those values of x, 0)\r\n" );
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document.write( "   f(x) = -3/11(x - 7)² - 9\r\n" );
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document.write( "      0 = -3/11(x - 7)² - 9\r\n" );
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document.write( "Multiply thru by 11\r\n" );
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document.write( "                    0 = -3(x - 7)² - 99\r\n" );
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document.write( "Divid through by -3\r\n" );
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document.write( "                    0 = (x - 7)² + 33\r\n" );
document.write( "                    0 = (x - 7)(x - 7) + 33\r\n" );
document.write( "                    0 = (x² - 14x + 49) + 33\r\n" );
document.write( "                    0 = x² - 14x + 49 + 33\r\n" );
document.write( "                    0 = x² - 14x + 82\r\n" );
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document.write( "We now find out if this has real or imaginary\r\n" );
document.write( "solutions by using the discriminant B²-4AC where\r\n" );
document.write( "A=4, B=-14, C=82\r\n" );
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document.write( "B²-4AC = (-14)²-4(4)(82) = -1116\r\n" );
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document.write( "When the discriminant is negative, the solutions\r\n" );
document.write( "are imaginary, so there are no x-intercepts.\r\n" );
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document.write( "Here is the graph\r\n" );
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document.write( "Here it is with the line of symmetry x = 7\r\n" );
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document.write( "Edwin
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