1. Table of Contents Rational Functions: Vertical Asymptotes Vertical Asymptotes: A vertical asymptote of a rational function is a vertical line (equation: x = number) such that as values of the independent variable, x, approach (get closer and closer to) the number (from either the left or right side of the number), the function values (y-values) decrease without bound or increase without bound. The next two slides illustrate the definition.

2. Table of Contents Rational Functions: Vertical Asymptotes Slide 2 The rational function shown graphed has vertical asymptote: x = 3, since as x-values approach 3 from the "left" of 3, the y values decrease without bound. -30 -20 -10 0 10 20 30 -2246 x2.72.82.92.99 y-18-28-58-598 x approaching 3 from the left y decreasing without bound

3. Table of Contents Rational Functions: Vertical Asymptotes Slide 3 Also note that as x-values approach 3 from the "right" of 3, the y values increase without bound. -30 -20 -10 0 10 20 30 -2246 x3.33.23.13.01 y223262602 x approaching 3 from the right y increasing without bound

4. Table of Contents Rational Functions: Vertical Asymptotes Slide 4 This function’s only vertical asymptote is x = 3. (This was the function shown graphed in the preceding slides!) Next, set the denominator equal to zero and solve. x – 3 = 0 x = 3 Example 1: Algebraically find the vertical asymptotes of, First, make sure the rational function can not be simplified further. In this case, f (x), is simplified.

5. Table of Contents Rational Functions: Vertical Asymptotes Slide 5 This function’s vertical asymptotes are x = - 2 and x = 1. Try: Algebraically find the vertical asymptotes of,

6. Table of Contents Rational Functions: Vertical Asymptotes