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document.write( "Hi, there--\r\n" );
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document.write( "THE PROBLEM:\r\n" );
document.write( "What is the graph of the rational function 
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document.write( "A SOLUTION:\r\n" );
document.write( "The graph of a rational function often has a restricted domain and asymptotes. \r\n" );
document.write( "Let's check for those.\r\n" );
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document.write( "DOMAIN:\r\n" );
document.write( "Because we are dividing one polynomial by another, we need to make sure we do not divide \r\n" );
document.write( "by zero. Set the denominator equal to zero, and solve for x.\r\n" );
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document.write( "Factor the left-hand side (difference of two squares).\r\n" );
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document.write( "Apply the Zero-Product Property which basically says, \"you are multiplying to factors together \r\n" );
document.write( "and the answer is zero. This means that at least one of those factors must equal zero too.\"\r\n" );
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document.write( "Either x + 3 = 0 or x - 3 = 0.\r\n" );
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document.write( "Solve for x.\r\n" );
document.write( "x = -3 or x = 3\r\n" );
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document.write( "In other words, when x = -3 or when x = 3, the denominator will equal 0, and we will be \r\n" );
document.write( "dividing by zero. We need to restrict the domain to keep his from happening.\r\n" );
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document.write( "Therefore, our domain is all real numbers, such that x is not equal to 3 or -3.\r\n" );
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document.write( "ASYMPTOTES:\r\n" );
document.write( "Most often, you will have vertical asymptotes at the zeroes of the denominator (x=-3 and \r\n" );
document.write( "x=3 in your problem.) \r\n" );
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document.write( "We want to make sure that the numerator and denominator do not have common factors, though.\r\n" );
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document.write( "Factor the numerator: 
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document.write( "Factor the denominator: 
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document.write( "Notice that (x-3) is a common factor in both the numerator and denominator. \r\n" );
document.write( "When this occurs we will not have a vertical asymptote at x=3, but we need to check the \r\n" );
document.write( "function at that point. We have\r\n" );
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document.write( "In this case, the graph will have a hole in the smooth curve right at x=3 because 0/0 is not a \r\n" );
document.write( "real number. \r\n" );
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document.write( "(NOTE: If you graph the function on a graphing calculator, you will not see this hole \r\n" );
document.write( "unless you REALLY zoom in. If you check the table of values, you will see ERROR for the \r\n" );
document.write( "y-value associated with x=3.)\r\n" );
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document.write( "When the degree of the polynomial in the numerator matches that of the polynomial in the \r\n" );
document.write( "denominator, we also have horizontal, not slant, asymptotes. To find these, divide the \r\n" );
document.write( "coefficient of the highest degree term in the numerator by the coefficient of the highest \r\n" );
document.write( "degree term in the denominator. In your problem, we have y=1/1 or y=1 as the horizontal \r\n" );
document.write( "asymptote. \r\n" );
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document.write( "X- AND Y-INTERCEPTS\r\n" );
document.write( "By solving for x = 0, you can find the y-intercept.\r\n" );
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document.write( "The graph crosses the y-axis at (0, -1/3).\r\n" );
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document.write( "By solving for y=0, you can find the x-inercepts. In this case y=0 if the numerator equals 0. Recall that we factored the numerator earlier.\r\n" );
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document.write( "Solving for x gives x=1 or x=3. The graph crosses the the x-axis at (1,0). (At x=3, we have the hole in the graph.)\r\n" );
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document.write( "Putting this all together, we have a graph with two branches: to the left of the vertical \r\n" );
document.write( "asymptote the graph resides above y=1. To the right of the vertical asymptote, the graph \r\n" );
document.write( "resides below y=1. Remember to mark the hole with at open circle and to indicate the \r\n" );
document.write( "intercepts.\r\n" );
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document.write( "Here is a (long) link to the graph of this function. You will need to cut and past it into your \r\n" );
document.write( "web browser:\r\n" );
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document.write( "http://graphsketch.com/?eqn1_color=1&eqn1_eqn=(x%5E2-4x%2B3)%2F(x%5E2-9)&eqn2_color=2&eqn2_eqn=&eqn3_color=3&eqn3_eqn=&eqn4_color=4&eqn4_eqn=&eqn5_color=5&eqn5_eqn=&eqn6_color=6&eqn6_eqn=&x_min=-17&x_max=17&y_min=-10.5&y_max=10.5&x_tick=1&y_tick=1&x_label_freq=5&y_label_freq=5&do_grid=0&do_grid=1&bold_labeled_lines=0&bold_labeled_lines=1&line_width=4&image_w=850&image_h=525\r\n" );
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document.write( "Hope this helps! Feel free to email if you have questions about the explanation.\r\n" );
document.write( "Mrs. Figgy\r\n" );
document.write( "math.in.the.vortex@gmail.com\r\n" );
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