Rational Functions Section 8.3 Day 2

Holes Holes are omitted points in a graph. In some cases, both the numerator and the denominator of a rational function will equal 0 for a particular value of x. A function is continuous with the exception for breaks. The difference between an asymptote and hole is: Holes are applied to cancelled expressions which cause the graph to be discontinuous For asymptote graphs, they generally approach the asymptote A hole CANNOT be a vertical asymptote A hole CANNOT be a zero HOLES TAKE PRECEDENCE OVER ASYMPTOTES 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 1 Identify any holes and vertical asymptotes of To identify any holes, factor the function out. 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 1 Identify any holes and vertical asymptotes of 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 1 Identify any holes and vertical asymptotes of What is leftover on bottom is the vertical asymptote 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 2 Graph, . Identify the zeros, y–intercept, vertical and horizontal asymptotes, and holes. Hole(s): Zero(s): Y–intercept: Vertical Asymptote: Horizontal Asymptote: x y –4 1/3 –3 –2 –1 Und 3 1 2 1.667 3/2 4 1.4 x y –4 –3 –2 –1 1 2 3 4 5/4/2019 12:17 PM 8.3: Graphing Rational Functions Day 2

8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 3 Graph, . Identify the zeros, y–intercept, vertical and horizontal asymptotes, and holes. x y –3 –2 –1 1 2 3 4 5 x y –3 –1.6 –2 –.75 –1 ½ 1 2 Und 3 8 4 7.5 5 Hole(s): Zero(s): Y–intercept: Vertical Asymptote: Horizontal Asymptote: 5/4/2019 12:17 PM 8.3: Graphing Rational Functions Day 2

8.3: Graphing Rational Functions Day 2 Your Turn Graph, . Identify the zeros, y–intercept, vertical and horizontal asymptotes, and holes. Hole(s): Zero(s): Y–intercept: Vertical Asymptote: Horizontal Asymptote: x y –3 –0.6 –2 –0. 5 –1 1/3 1 2 Und 3 4 x y –3 –2 –1 1 2 3 4 5/4/2019 12:17 PM 8.3: Graphing Rational Functions Day 2

8.3: Graphing Rational Functions Day 2 Example 3 Write an equation where there is a hole at (2, 5/3), zero is at (–3, 0), vertical asymptote is at x = –1, and horizontal asymptote is at y = 1. 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 4 Write an equation where there is a hole at (–2/3, –7/13), zero is at (1/2, 0), vertical asymptote is at x = –5, and horizontal asymptote is at y = 2. 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Example 5 Write an equation where there is a hole at (1, 1), zero is at (0, 0), vertical asymptote is at x = –1, and horizontal asymptote is at y = 2. 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Your Turn Write a rational equation where the zero is at (-5, 0), a hole exists at (2, 14/11), vertical asymptote is at x = –7/2, horizontal asymptote is at y = 1, and the y-intercept is at (0, 10/7). 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM

8.3: Graphing Rational Functions Day 2 Assignment WKST 2 (double homework) 8.3: Graphing Rational Functions Day 2 5/4/2019 12:17 PM