Final answer:
When graphing a rational function, identify the VAs by setting the denominator to zero, the HAs by comparing the degrees of the numerator and denominator coefficients, holes by finding common factors, x-intercepts by solving when y=0, and y-intercepts by solving when x=0.
Step-by-step explanation:
When graphing rational functions, it's important to identify several key features: vertical asymptotes (VAs), horizontal asymptotes (HAs), holes, x-intercepts (x-int), and y-intercepts (y-int). Vertical asymptotes occur where the denominator of the rational function is zero, but the numerator is not zero at the same point. To find them, set the denominator equal to zero and solve for x.
Horizontal asymptotes exist when x approaches infinity, indicating the function's behavior as it extends far to the left or right. To find the HA, compare the degrees of the numerator and denominator. If they are the same, the HA is the ratio of the leading coefficients. If the degree of the numerator is less than the denominator, the HA is y=0. If the degree of the numerator is greater, there isn't a HA, but there might be an oblique asymptote.
Holes occur when there is a common factor in both the numerator and the denominator that cancels out. To identify holes, factor both the numerator and denominator fully and look for common factors. The x-value of the hole is the solution to setting the common factor equal to zero.
To find the x-intercept(s), set y equal to zero and solve for x. For the y-intercept, set x equal to zero and solve for y. These intercepts represent where the graph crosses the axes. Graphing each of these elements accurately can help illustrate the overall shape and behavior of the rational function on a coordinate plane.