Asymptotes Rise Their Lovely Heads Rational Functions Asymptotes Rise Their Lovely Heads

Definition of a Rational Function Domain – CANNOT DIVIDE BY ZERO Function in lowest terms – cancelling functions that are similar in the numerator and denominator

Graphing y = 1/x^2 Why does it look like this? What is the transformation?

Transformations Review H(x) = 1/(x – 2) ^2 + 1 (x – 2) moves the graph to the right 2 + 1 moves the graph up 1 Now graph the y = 1/x^2 graph

Asymptotes Definition Three types: A. Vertical Asymptotes B. Horizontal Asymptotes C. Oblique Asymptotes

Vertical Asymptotes To find vertical asymptotes write the function in lowest terms Set the denominator equal to zero The asymptote will be located at x = this number

Horizontal Asymptotes Rules for finding horizontal asymptotes: 1.If the numerator and denominator have the same degree. 2. If the numerator is a smaller degree than the denominator 3. If the numerator is a larger degree than the denominator

Horizontal Asymptote The equation of the asymptote is y = the leading coefficients The equation of the asymptote is y = 0 There is no horizontal asymptote

Oblique Asymptotes If the degree of the numerator is larger than the degree of the denominator then the graph has an oblique asymptote. Divide the numerator into the denominator

Summary See page 223 for all of the rules.

Graphs Step One: Find the domain of the rational function Step Two: Write the function in lowest terms Step Three: Locate the intercepts of the graph Step Four: Locate the vertical asymptotes Step Five: Locate the horizontal or oblique asymptotes

Graphs Step 6: Identify the behavior of the graph around the vertical asymptotes

Graphs Example of function with oblique asymptote More Examples

Constructing a Rational Function Given Asymptotes and Intercepts Example

Constructing a Rational Function from Its Graph

Applications P. 235 problem 46 & 48