Section 2.6 Rational Functions and Their Graphs

Rational Function are quotients of polynomial functions. This means that rational functions can be expressed as The domain of a rational function is the set of all real numbers EXCEPT the x-values that make the denominator zero. Rational Functions

Example 1 Find the domain of the rational function.

Example 2

The Force Fields Of Mathematics Vertical Asymptotes of Rational Functions

Here’s another one to examine.

Arrow Notation (Intro to Calculus notation)

One of these things is not like the other. What could it be?

Find the vertical asymptote, if any, of the graph of the rational function. Example 3

Find the vertical asymptote, if any, of the graph of the rational function. Example 4

Find the vertical asymptote, if any, of the graph of the rational function. Example 5 Removed

Removable Domains and Asymptotes!!!! Consider the function The domain is… What happens to the domain and asymptote? However, we can simplify the rational expression... A graph with a hole corresponding to the denominator’s zero. Your calculator will not show the hole, but you must on your papers.

Horizontal Asymptotes of Rational Functions

You must memorize these!!!

One of these things is not like the other! Which one could it be? Notice how the horizontal asymptote intersects the graph.

Find the horizontal asymptote, if any, of the graph of the rational function. Example 6

Find the horizontal asymptote, if any, of the graph of the rational function. Example 7

Example 8

Example 9

The graph of a rational function has a slant asymptote if the degree of the numerator is ONE more than the degree of the denominator. The equation of the SLANT ASYMPTOTE can be found by division. The equation of the slant asymptote is the quotient with the remainder dropped. Slant Asymptotes

Example 10

Example 11

(a) (b) (c) (d)

(a) (b) (c) (d)

(a) (b) (c) (d)

Using Transformations to Graph Rational Functions Graphs of Common Rational Functions

Transformations of Rational Functions

Example 12

Example 13

Graphing Calculator – TI 84 Connected mode Dot Mode

Example 14

The Fort Myers Fishing Company discovered a better material for making fishing reels. The fixed monthly cost is $10,000 for the cost of rental of space, manufacturing equipment, as well as wages and benefits for it’s employees. It costs $10 for materials to make each fishing reel.