document.write( "Question 39513: How do you compute the intercepts of a quadratic function?\r
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Algebra.Com's Answer #24955 by AnlytcPhil(1806) $\"\"$ $\"About$
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How do you compute the intercepts of a quadratic function? \r\n" );
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document.write( "Intercepts are points where a graph crosses the x axis or the y axis.\r\n" );
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document.write( "Every point on the x-axis has 0 for its y-coordinate.  And vice-versa, that is,\r\n" );
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document.write( "every point on the y-axis has 0 for its x-coordinate.\r\n" );
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document.write( "A quadratic graph  \r\n" );
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document.write( "1. Is either shaped like a U or an upside down U.\r\n" );
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document.write( "  or  \r\n" );
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document.write( "2. Always has exactly 1 y-intercept\r\n" );
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document.write( "3. Sometimes has 2 x-intercepts, like the two graphed above\r\n" );
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document.write( "4.  Sometimes has NO x-intercepts, like this one just below:\r\n" );
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document.write( "5. Sometimes has exactly ONE x-intercept, like this one below\r\n" );
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document.write( "6. Has as its general equation\r\n" );
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document.write( "     y = Ax² + Bx + C\r\n" );
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document.write( "     where A, B and C can represent positive or negative numbers,\r\n" );
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document.write( "7.  B and/or C is sometimes 0, but A is never 0. \r\n" );
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document.write( "How to find the y-intercept:\r\n" );
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document.write( "Substitute 0 for x and solve for y:\r\n" );
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document.write( "Then the y-intercept is\r\n" );
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document.write( "(0, whatever you got)\r\n" );
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document.write( "How to find the x-intercepts:\r\n" );
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document.write( "Substitute 0 for y and solve for x:\r\n" );
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document.write( "Then the x-intercept is (whatever you got, 0)\r\n" );
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document.write( "Example 1:  \r\n" );
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document.write( "y = x² + 2x - 3\r\n" );
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document.write( "To find its y-intercept, substitute 0 for x, and solve for y\r\n" );
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document.write( "y = 0² + 2(0) - 3\r\n" );
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document.write( "y = -3\r\n" );
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document.write( "So the y-intercept is the point (0, -3)\r\n" );
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document.write( "To find its x-intercepts, substitute 0 for y, and solve for x\r\n" );
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document.write( "0 = x² + 2x - 3\r\n" );
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document.write( "x² + 2x - 3 = 0\r\n" );
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document.write( "(x + 3)(x - 1) = 0\r\n" );
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document.write( "x = -3, and x = 1\r\n" );
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document.write( "So the two x-intercepts are (-3, 0) and (1, 0)\r\n" );
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document.write( "Its graph looks like this\r\n" );
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document.write( "Example 2:\r\n" );
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document.write( "f(x) = -x² + x - 1\r\n" );
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document.write( "Replace f(x) by y\r\n" );
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document.write( "y = -x² + x - 1\r\n" );
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document.write( "To find its y-intercept, substitute 0 for x, and solve for y\r\n" );
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document.write( "y = -0² + 0 - 1\r\n" );
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document.write( "y = -1\r\n" );
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document.write( "So the y-intercept is the point (0, 1)\r\n" );
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document.write( "To find its x-intercepts, substitute 0 for y, and solve for x\r\n" );
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document.write( "0 = -x² + x - 1\r\n" );
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document.write( "x² - x + 1 = 0\r\n" );
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document.write( "That doesn't factor, so we use the\r\n" );
document.write( "quadratic equation and find\r\n" );
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document.write( "x = 1/2 ± Ö3/2i\r\n" );
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document.write( "which are imaginary, so there are no x-intercepts.\r\n" );
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document.write( "Its graph looks like this\r\n" );
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document.write( "Example 3:  \r\n" );
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document.write( "y = x² + 6x + 9\r\n" );
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document.write( "To find its y-intercept, substitute 0 for x, and solve for y\r\n" );
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document.write( "y = 0² + 6(0) + 9\r\n" );
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document.write( "y = 9\r\n" );
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document.write( "So the y-intercept is the point (0, 9)\r\n" );
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document.write( "To find its x-intercepts, substitute 0 for y, and solve for x\r\n" );
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document.write( "0 = x² + 6x + 9\r\n" );
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document.write( "x² + 6x + 9 = 0\r\n" );
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document.write( "(x + 3)(x + 3) = 0\r\n" );
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document.write( "x = -3, and x = -3\r\n" );
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document.write( "These are the same so there is only one x-intercept\r\n" );
document.write( "(-3, 0)\r\n" );
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document.write( "Its graph looks like this\r\n" );
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document.write( "It just \"sits\" on the x-axis at one point (-3, 0)\r\n" );
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document.write( "Edwin
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