Sign (intervals of definition, increasing, decreasing)
The graph of a rational function can be broken into disconnected segments by the presence of vertical asymptotes. Remember, at a vertical asymptote the function approaches either positive or negative infinity (sometimes both). To graph a rational function correctly it helps to determine where it is positive or negative.
Here is an organized way to do this.
A. Find the zeros and undefined x-values of the function (places where numerator or denominator are zero).
B. Arrange these values in increasing order.
C. Divide the x-axis into intervals around these values. For example, if the values are -1 and 0, the intervals are:
-1 < x, -1 < x < 0, and 0 < x. If there are N values, there will be N+1 intervals.
D. Test the sign of the function on each interval. You don’t have to compute exact values of the function, just check to see if it is positive or negative.
Here is a set of problems that guide through the process:
1. f(x) = (x2 + x) / (x2 + 6x + 5)
Factor the numerator and denominator of f(x).
2. Where does f(x) have zero value(s)?
3. Where does f(x) have vertical asymptote(s)?
4. Where does f(x) have a hole?
5. Using the information from problems 1-4, test the sign of f(x) over the appropriate intervals.