Could someone please help me with this.\r\n" );
document.write( "Could someone show me how to sketch a function that is odd, one that\r\n" );
document.write( "is even, and one that is neither. \r\n" );
document.write( "Please and thank you!!!! \r\n" );
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document.write( "Here are some that are EVEN. What's on the left side of the y-axis \r\n" );
document.write( "is a mirror image of what is on the right side of the y-axis. They \r\n" );
document.write( "are said to be symmetrical with respect to the y-axis:\r\n" );
document.write( "EVEN function:\r\n" );
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\r\n" );
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document.write( "EVEN function:\r\n" );
document.write( "
\r\n" );
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document.write( "EVEN function:\r\n" );
document.write( "
\r\n" );
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document.write( "EVEN function:\r\n" );
document.write( "
\r\n" );
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document.write( "Here are some that are odd. What's on the left side of the y-axis is\r\n" );
document.write( "upside down from what is on the right side of the y-axis.\r\n" );
document.write( "They are said to be symmetical with respect to the ORIGIN.\r\n" );
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document.write( "Notice that an ODD function's graph looks exactly the same upside-down \r\n" );
document.write( "as it does right-side up.\r\n" );
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document.write( "ODD function:\r\n" );
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\r\n" );
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document.write( "ODD function:\r\n" );
document.write( "
\r\n" );
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document.write( "ODD function:\r\n" );
document.write( "
\r\n" );
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document.write( "ODD function:\r\n" );
document.write( "
\r\n" );
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document.write( "Here is one that is neither. \r\n" );
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document.write( "Neither ODD nor EVEN\r\n" );
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document.write( "Incidentally there is only one function that is both ODD and\r\n" );
document.write( "EVEN, and that is the function whose graph is the x-axis. Its\r\n" );
document.write( "equation is y = 0, graphed in purple below:\r\n" );
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document.write( "
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document.write( "-------------------------------------------------------\r\n" );
document.write( "One more that I am having trouble with is Graph y=2x^2-4x-6. Label \r\n" );
document.write( "all x-intercepts, y-intercepts, and the vertex. \r\n" );
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document.write( "Here is the graph. You can find it by finding and plotting some points, \r\n" );
document.write( "or, what would be better, just wait until you get the points you are \r\n" );
document.write( "asked to find below. But here it is in advance:\r\n" );
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\r\n" );
document.write( "Notice that it is neither an EVEN nor an ODD function.\r\n" );
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document.write( "To find the x-intercepts, replace y by 0 and solve for x:\r\n" );
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document.write( "y = 2x² - 4x - 6\r\n" );
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document.write( "0 = 2x² - 4x - 6\r\n" );
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document.write( "Divide every term by 2\r\n" );
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document.write( "0/2 = 2x²/2 - 4x/2 - 6/2\r\n" );
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document.write( "0 = x² - 2x - 3\r\n" );
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document.write( "Factor the right side:\r\n" );
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document.write( "0 = (x + 1`)(x - 3)\r\n" );
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document.write( "Set each factor = 0:\r\n" );
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document.write( "x + 1 = 0 gives x = -1\r\n" );
document.write( "x - 3 = 0 gives x = 3\r\n" );
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document.write( "So the x-intercepts are (-1,0) and (3,0).\r\n" );
document.write( "They are the two points where the graph \r\n" );
document.write( "crosses the x-axis:\r\n" );
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document.write( "To find the y-intercept, replace x by 0 and solve for y\r\n" );
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document.write( "y = 2x² - 4x - 6 \r\n" );
document.write( "y = 2(0)² - 4(0) - 6\r\n" );
document.write( "y = 0 - 0 - 6\r\n" );
document.write( "y = -6\r\n" );
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document.write( "So the y-intercept is (0,-6). It is the point at which\r\n" );
document.write( "the graph crosses the y-axis.\r\n" );
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document.write( "To find the vertex:\r\n" );
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document.write( "Factor the coefficient 2 out of the first two terms:\r\n" );
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document.write( "y = 2(x² - 2x) - 6\r\n" );
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document.write( "Complete the square by \r\n" );
document.write( "(1) multiplying the coefficient of x, -2, inside the parentheses\r\n" );
document.write( " by 1/2, ( -2·1/2 = -1, then\r\n" );
document.write( "(2) squaring this result, (-1)² = 1\r\n" );
document.write( "(3) adding and subtracting this inside the parentheses:\r\n" );
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document.write( "y = 2(x² - 2x + 1 - 1) - 6\r\n" );
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document.write( "Change the parentheses to brackets since they will soon have to\r\n" );
document.write( "hold parentheses inside:\r\n" );
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document.write( "y = 2[x² - 2x + 1 - 1] - 6\r\n" );
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document.write( "Factor the first three terms as (x-1)(x-1) inside the brackets:\r\n" );
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document.write( "y = 2[(x - 1)(x - 1) - 1] - 6\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( "y = 2[(x - 1)² - 1] - 6\r\n" );
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document.write( "Remove the brackets, keeping the (x-1)² intact:\r\n" );
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document.write( "y = 2(x - 1)² - 2 - 6\r\n" );
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document.write( "Combine the -2 and the -6\r\n" );
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document.write( "y = 2(x - 1)² - 8\r\n" );
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document.write( "Compare that to\r\n" );
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document.write( "y = a(x - h)² + k\r\n" );
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document.write( "and see that a = 2, h = 1, k = -8\r\n" );
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document.write( "So the vertex is (h,k) = (1,-8).\r\n" );
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document.write( "The vertex is the very bottom point on this\r\n" );
document.write( "graph. (Sometimes it's the very top point,\r\n" );
document.write( "It's always the \"turning point\")\r\n" );
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document.write( "Edwin
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document.write( "![\"\"](\"/images/stars/stars-13.gif\")
