Sign (intervals of definition, increasing, decreasing)<\/p>\n
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.<\/p>\n
Here is an organized way to do this.<\/p>\n
A. Find the zeros and undefined x-values of the function (places where numerator or denominator are zero).
\nB. Arrange these values in increasing order.
\nC. Divide the x-axis into intervals around these values. For example, if the values are -1 and 0, the intervals are:
\n-1 < x, -1 < x < 0, and 0 < x. If there are N values, there will be N+1 intervals.
\nD. 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.<\/p>\n
Here is a set of problems that guide through the process:<\/p>\n
1. f(x) = (x2<\/sup> + x) \/ (x2<\/sup> + 6x + 5) 2. Where does f(x) have zero value(s)?<\/p>\n 3. Where does f(x) have vertical asymptote(s)?<\/p>\n 4. Where does f(x) have a hole?<\/p>\n 5. Using the information from problems 1-4, test the sign of f(x) over the appropriate intervals.<\/p>\n
\nFactor the numerator and denominator of f(x).<\/p>\n