College Algebra Chapter 3 Polynomial and Rational Functions Section 3.5 Rational Functions

Concepts 1. Apply Notation Describing Infinite Behavior of a Function 2. Identify Vertical Asymptotes 3. Identify Horizontal Asymptotes 4. Identify Slant Asymptotes 5. Graph Rational Functions 6. Use Rational Functions in Applications

Definition of a Rational Function Let p(x) and q(x) be polynomials where q(x) ≠ 0. A function f defined by is called a rational function.

Notation Describing Infinite Behavior of a Function NotationMeaning x approaches c from the right (but will not equal c). x approaches c from the left (but will not equal c). x approaches infinity (x increases without bound). x approaches negative infinity (x decreases without bound).

Example 1: The graph of is given. Complete the statements.

Concepts 1. Apply Notation Describing Infinite Behavior of a Function 2. Identify Vertical Asymptotes 3. Identify Horizontal Asymptotes 4. Identify Slant Asymptotes 5. Graph Rational Functions 6. Use Rational Functions in Applications

Definition of a Vertical Asymptote The line x = c is a vertical asymptote of the graph of a function f if f (x) approaches infinity or negative infinity as x approaches c from either side.

Definition of a Vertical Asymptote, continued

Identifying Vertical Asymptotes of a Rational Function Let where p(x) and q(x) have no common factors other than 1. To locate the vertical asymptotes of, determine the real numbers x where the denominator is zero, but the numerator is nonzero.

Example 2: Identify the vertical asymptotes (if any).

Example 3: Identify the vertical asymptotes (if any).

Example 4: Identify the vertical asymptotes (if any).

Concepts 1. Apply Notation Describing Infinite Behavior of a Function 2. Identify Vertical Asymptotes 3. Identify Horizontal Asymptotes 4. Identify Slant Asymptotes 5. Graph Rational Functions 6. Use Rational Functions in Applications

Definition of a Horizontal Asymptote The line y = d is a horizontal asymptote of the graph of a function f if f (x) approaches d as x approaches infinity or negative infinity.

Definition of a Horizontal Asymptote, continued

Let f be a rational function defined by where n is the degree of the numerator and m is the degree of the denominator. 1.If n > m, f has no horizontal asymptote. 2.If n < m, then the line y = 0 (the x-axis) is the horizontal asymptote of f. 3.If n = m, then the line is the horizontal asymptote of f. Identifying Horizontal Asymptotes of a Rational Function

Example 5: Identify the horizontal asymptotes (if any).

Example 6: Identify the horizontal asymptotes (if any).

Example 7: Identify the horizontal asymptotes (if any).

The graph of a rational function may not cross a vertical asymptote. The graph may cross a horizontal asymptote. Crossing Horizontal Asymptotes

Example 8: Determine if the function crosses its horizontal asymptote.

Concepts 1. Apply Notation Describing Infinite Behavior of a Function 2. Identify Vertical Asymptotes 3. Identify Horizontal Asymptotes 4. Identify Slant Asymptotes 5. Graph Rational Functions 6. Use Rational Functions in Applications

A rational function will have a slant asymptote if the degree of the numerator is exactly one greater than the degree of the denominator. To find an equation of a slant asymptote, divide the numerator of the function by the denominator. The quotient will be linear and the slant asymptote will be of the form y = quotient. Identifying Slant Asymptotes of a Rational Function

Example 9: Determine the slant asymptote of.

Concepts 1. Apply Notation Describing Infinite Behavior of a Function 2. Identify Vertical Asymptotes 3. Identify Horizontal Asymptotes 4. Identify Slant Asymptotes 5. Graph Rational Functions 6. Use Rational Functions in Applications

where p(x) and q(x) are polynomials with no common factors. Graphing a Rational Function Intercepts:Determine the x- and y-intercepts Asymptotes:Determine if the function has vertical asymptotes and graph them as dashed lines. Determine if the function has a horizontal asymptote or a slant asymptote, and graph the asymptote as a dashed line. Determine if the function crosses the horizontal or slant asymptote.

Graphing a Rational Function, continued Symmetry:f is symmetric to the y-axis if f (–x) = f (x). f is symmetric to the origin if f (–x) = –f (x). Plot points:Plot at least one point on the intervals defined by the x-intercepts, vertical asymptotes, and points where the function crosses a horizontal or slant asymptote.

Example 10: Graph.

Example 10 continued:

Example 11: Graph.

Example 11 continued:

Example 12: Graph.

Example 12 continued:

Example 13: Use transformations to graph.

Concepts 1. Apply Notation Describing Infinite Behavior of a Function 2. Identify Vertical Asymptotes 3. Identify Horizontal Asymptotes 4. Identify Slant Asymptotes 5. Graph Rational Functions 6. Use Rational Functions in Applications

Example 14: Young's rule is a formula for calculating the approximate dosage for a child, given the adult dosage. For an adult dosage of d mg, the child's dosage C(x) may be calculated by x is the child's age in years and.

Example 14 continued: a.What is the horizontal asymptote of the function? What does it mean in the context of this problem? b.What is the vertical asymptote of the function?

Example 14 continued: c.For an adult dosage of 50 mg, what is the approximate dosage for a 9-year-old child? Graph: