document.write( "Question 593562: Please help me graph the rational function y = $\"4x%2F%28x%5E3-4x%29\"$ . \n" ); document.write( "
Algebra.Com's Answer #376366 by Edwin McCravy(20060) $\"\"$ $\"About$
You can put this solution on YOUR website!
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document.write( "y = \r\n" );
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document.write( "Factor the denominator:\r\n" );
document.write( "Factor out x in the denominator:\r\n" );
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document.write( "y = \r\n" );
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document.write( "Notice that x is a factor of both the numerator and denominator.  In earlier\r\n" );
document.write( "algebra courses you were told just to cancel the like factors in the numerator\r\n" );
document.write( "and denominator.  However they told you that because they didn't want to get\r\n" );
document.write( "you too involved in those earlier courses.\r\n" );
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document.write( "What they didn't tell you in those earlier courses is that if you cancel the\r\n" );
document.write( "like factors in the numerator and denominator, you must NOT INCLUDE on your\r\n" );
document.write( "graph any point of the resulting equation's graph after canceling them.  So\r\n" );
document.write( "what we do in this course is to cancel, draw the graph, then remove any \r\n" );
document.write( "point(s) for which the canceled factors equals 0.\r\n" );
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document.write( "First we cancel the x's, but we must not forget that we canceled them as we did\r\n" );
document.write( "in earlier courses:\r\n" );
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document.write( "y = \r\n" );
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document.write( "y = \r\n" );
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document.write( "Draw a set of axes:\r\n" );
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document.write( "Find the vertical asymptotes:\r\n" );
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document.write( "Set the denominator = 0\r\n" );
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document.write( "        x² - 4 = 0\r\n" );
document.write( "(x - 2)(x + 2) = 0\r\n" );
document.write( "x - 2 = 0;  x + 2 =  0\r\n" );
document.write( "    x = 2;      x = -2\r\n" );
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document.write( "So there are two vertical asymptotes:\r\n" );
document.write( "x = 2 (a vertical line through 2 on the x axis)\r\n" );
document.write( "x = -2 (a vertical line through -2 on the x axis)\r\n" );
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document.write( "When the degree of the numerator is less than the degree of the denominator,\r\n" );
document.write( "the horizontal asymptote is the x-axis, whose equation is y=0.\r\n" );
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document.write( "I will color them all the asymptotes green, but on your paper you can draw them\r\n" );
document.write( "as dotted lines:\r\n" );
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document.write( "We can if we like check for symmetry with respect to the y-axis and origin by\r\n" );
document.write( "replacing x by (-x), simplifying and seeing whether we get:\r\n" );
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document.write( "1. the original equation, the original equation, which would mean the graph\r\n" );
document.write( "   would be symmetrical with respect to the y-axis.\r\n" );
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document.write( "2. the original equation except with the right side multiplied by -1, which\r\n" );
document.write( "   would mean the graph is symmetrical with respect to the origin.\r\n" );
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document.write( "3. Neither, in which case the graph would not have symmetry with respect to the\r\n" );
document.write( "   y-axis or the origin.\r\n" );
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document.write( "y = \r\n" );
document.write( "y = \r\n" );
document.write( "y = \r\n" );
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document.write( "It came back to the y = , so it is symmetrical with respect to\r\n" );
document.write( "the y-axis.\r\n" );
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document.write( "So we only need to get a few points on the right left and right of the vertical\r\n" );
document.write( "asymptote:\r\n" );
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document.write( "  x |  y\r\n" );
document.write( "  0 | -1  \r\n" );
document.write( "  1 | -1.3\r\n" );
document.write( "1.5 | -2.3\r\n" );
document.write( "2.5 |  1.8\r\n" );
document.write( "  3 |  0.8\r\n" );
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document.write( "We plot those:\r\n" );
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document.write( "Then due to symmetry, match them on the left side of the y-axis:\r\n" );
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document.write( "Draw in the graph:\r\n" );
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document.write( "But we cannot stop here as we could have in lower courses of algebra, because\r\n" );
document.write( "the graph of y =  is NOT quite the same graph as the graph of\r\n" );
document.write( "y = .\r\n" );
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document.write( "We must remove from the graph of y =  any point for which the\r\n" );
document.write( "factor we canceled equals 0. So we must remove the point when x = 0, which is\r\n" );
document.write( "the point (0,-1).\r\n" );
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document.write( "So the graph of y =  is this\r\n" );
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document.write( "Notice the circle around the point (0,-1).  The graph does not include that\r\n" );
document.write( "point.  So we say \"there is a removeable discontinuity at x=0\".  Colloquially\r\n" );
document.write( "we say \"There is a hole in the graph at x=0\". \r\n" );
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document.write( "Edwin
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