Question 1113853
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Graph the rational function:
{{{r(x)=((x-4)(x+1))/((x-1)(x+4))}}}<br>
(1) zeros: The function value is 0 whenever the numerator is 0 -- when x=4 or x=-1.
(2) vertical asymptotes: The function is undefined and so has a vertical asymptote whenever the denominator is 0 -- when x=1 or x=-4.
(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.<br>
(4) intervals where the function value is positive or negative:<br>
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.<br>
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.<br>
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.
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.
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.<br>
(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
{{{((x-4)(x+1))/((x-1)(x+4))=1}}}
{{{(x-4)(x+1) = (x-1)(x+4)}}}
{{{x^2-3x-4 = x^2+3x-4}}}
{{{6x=0}}}
{{{x=0}}}<br>
The function value crosses the horizontal asymptote at x=0, and only there.<br>
With all of the above, the basic behavior of the function is determined.<br>
Here is a graph...:<br>
{{{graph(400,400,-10,10,-10,10,((x-4)(x+1))/((x-1)(x+4)))}}}