Problem 113 - College Algebra

Question

Describe how to graph a rational function.

Step-by-Step Solution

Answer

To graph a rational function, determine the horizontal and vertical asymptotes, find the x and y intercepts, identify any potential holes and then plot these onto the graph. After that, gradually sketch the function properly.

Step 1: Identifying the Horizontal Asymptote

The horizontal asymptote of a rational function can be found by examining the degrees of the polynomials in the numerator and the denominator. If the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, then the horizontal asymptote is y = 0. If the degrees are equal, you divide the leading coefficients of the numerator and denominator. If the degree of the polynomial in the numerator is greater than the denominator, there is no horizontal asymptote.

Step 2: Identifying the Vertical Asymptotes

The vertical asymptotes occur at the x-values that make the denominator equal to zero. In other words, you find the vertical asymptotes by setting the denominator equal to zero and solving for x. Each solution will be a vertical asymptote. If the solution also makes the numerator zero, then it is not a vertical asymptote, but a hole.

Step 3: Identifying the x and y-intercepts

The x-intercept(s) can be found by setting the entire function equal to zero, and solving for x. To find the y-intercept (there's only one), simply plug 0 in for x.

Step 4: Plotting the Graph

Using the x and y-intercepts, asymptotes and possible holes identified in the earlier steps, sketch the graph carefully. Other points can be obtained by substituting x-values into the function. Ensure that your graph is asymptotic to the vertical and horizontal asymptotes you found in steps 1 and 2.