![\"\"](\"/images/stars/stars-13.gif\")
You can put this solution on YOUR website!
\n" ); document.write( "Graph the rational function:
\n" ); document.write( "
\n" ); document.write( "(1) zeros: The function value is 0 whenever the numerator is 0 -- when x=4 or x=-1.
\n" ); document.write( "(2) vertical asymptotes: The function is undefined and so has a vertical asymptote whenever the denominator is 0 -- when x=1 or x=-4.
\n" ); document.write( "(3) horizontal asymptote: As x gets very large (positive or negative) the constants become insignificant; the function value approaches x^2/x^2 = 1; the horizontal asymptote is y=1.
\n" ); document.write( "(4) intervals where the function value is positive or negative:
\n" ); document.write( "The x values -4, -1, 1, and 4 where the numerator or denominator is zero break the x axis into intervals; we need to determine whether the function value is positive or negative in each interval.
\n" ); document.write( "Most sources I have seen will tell you to use a test value in each interval. I find that process too repetitive; you can accomplish the same thing with less work if you simply imagine \"walking\" along the x axis and see what happens to the sign of the function value each time you cross from one of the intervals to the next.
\n" ); document.write( "For this problem, we can see that all 4 factors in the numerator and denominator are negative for large negative values of x; that means the function value is positive from negative infinity to x=-4.
\n" ); document.write( "Then when we pass x=-4, the sign of one factor changes; that changes the sign of the function value. So when we pass x=-4, the function value becomes negative.
\n" ); document.write( "Similarly, the function value changes from negative to positive when we pass x=-1; from positive to negative when we pass x=1, and finally from negative to positive when we pass x=4.
\n" ); document.write( "(5) Last we should find the values of x, if any, where the graph of the function crosses the horizontal asymptote. To do that, since the horizontal asymptote is y=1, we need to find all the solutions to
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "The function value crosses the horizontal asymptote at x=0, and only there.
\n" ); document.write( "With all of the above, the basic behavior of the function is determined.
\n" ); document.write( "Here is a graph...:
\n" ); document.write( "