Knowing where the function f(x) is positive and where is it negative is of great value in sketching its graph.    Polynomial functions  could only change sign as the graph crossed the x axis.  With rational functions, the function may have different signs on either side of the zeros of the denominator as well.   We therefore have to use  the zeros of the denominator as well as zeros of the numerator to partition the domain of the function into intervals and determine the sign of the function in each interval.  For example, if  ,  where x = -2 is a zero of the numerator, and x = 2 is a zero of the denominator, we consider  the regions  x < -2,  -2 < x < 2,  and x > 2.

Pick any convenient value for x < -2, say  x = -3, evaluate f(-3) =  1/5 > 0, so the graph lies above the x-axis  for x < -2.
Pick a value for x between -2 and 2,  say  x = 0, and evaluate f(0) = -1 < 0, so the graph lies below the axis for -2 < x < 2.
Pick a value for x > 2, say x = 3, and evaluate f(3) = 5 > 0, so the graph lies above the axis for x > 2.

Similarly, if   where the set of roots of the numerator and denominator is {-2,0,2}, we consider the regions x<-2,  -2<x<0,  and  0<x<2.

Pick any convenient value for x < -2, say  x = -3, evaluate f(-3) =  (-5)/(+15)  < 0, so the graph lies below the x-axis  for x < -2.
Pick a value for x between -2 and 0,  say  x = -1, and evaluate f(-1) = (-3)/(-1) > 0, so the graph lies above the axis for -2 < x < 0.
Pick a value for x between 0 and 2,  say  x = 1, and evaluate f(1) = (-1)/(3) < 0, so the graph lies below the axis for 0 < x < 2.
Pick a value for x > 2, say x = 3, and evaluate f(3) =1/15 > 0, so the graph lies above the axis for x > 2.
We can now summarize the second step in drawing the graph of a rational function:
 

STEP 2  Determine where the  function is negative and where positive.

 

To continue, click here to return to the overall discussion of graphing rational functions.