Rational Functions Vertical Asymptotes A vertical asymptote of

Rational Functions: 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. Table of Contents

Rational Functions: Vertical Asymptotes 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. -2 30 20 10 0 2 4 6 -10 -20 -30 x 2. 7 2. 8 2. 99 y -18 -28 -598 Table of Contents x approaching 3 from the left y decreasing without bound Slide 2

Rational Functions: Vertical Asymptotes Also note that as x-values approach 3 from the "right" of 3, the y values increase without bound. -2 30 20 10 0 2 4 6 -10 -20 -30 x 3. 3 3. 2 3. 1 3. 01 y 22 32 62 602 Table of Contents x approaching 3 from the right y increasing without bound Slide 3

Rational Functions: Vertical Asymptotes 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. Next, set the denominator equal to zero and solve. x – 3 = 0 x=3 This function’s only vertical asymptote is x = 3. (This was the function shown graphed in the preceding slides!) Table of Contents Slide 4

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

Rational Functions: Vertical Asymptotes Table of Contents