Rational Functions

Introduction to Rational Functions

Rational functions are a fundamental topic in College Algebra, representing functions that are ratios of polynomials. Understanding their properties, domain, and behavior is essential for analyzing and graphing these functions.

• Definition: A rational function is any function that can be written as , where and are polynomials and .

• Domain: The domain of a rational function is all real numbers except those for which the denominator equals zero.

• Lowest Terms: To write a rational function in lowest terms, factor both numerator and denominator and cancel any common factors.

• Procedure:

  1. Find the domain by setting the denominator equal to zero and solving for .

  2. Factor numerator and denominator.

  3. Cancel common factors to simplify.

• Example: Factor denominator: Cancel common factor : , with domain

Practice Problems

• Find the domain and write in lowest terms: Domain:

• Domain: , so

Asymptotes of Rational Functions

Introduction to Asymptotes

Asymptotes are lines that a graph approaches but never touches. Rational functions may have vertical, horizontal, or slant asymptotes, which describe their behavior as approaches certain values or infinity.

• Vertical Asymptotes (VA): Occur where the denominator equals zero (after simplifying).

• Horizontal Asymptotes (HA): Describe the end behavior as .

• Removable Discontinuities (Holes): Occur where a common factor cancels in both numerator and denominator.

• Example Table: Values of for various show the function approaches zero as and is undefined at .

Types of Asymptotes

• Vertical Asymptotes:

  • Set denominator equal to zero and solve for .

  • Example: After canceling , VA at .

• Horizontal Asymptotes:

  • Compare degrees of numerator and denominator:

  • If degree numerator < degree denominator:

  • If degree numerator = degree denominator:

  • If degree numerator > degree denominator: No horizontal asymptote (may be a slant asymptote).

• Removable Discontinuities (Holes):

  • Factor numerator and denominator.

  • Set common factor equal to zero and solve for .

  • Example: , so hole at .

Summary Table: Asymptote Types

Graphing Rational Functions

Graphing Using Transformations

Many rational functions can be graphed by applying transformations to basic parent functions such as or . Transformations include shifts, reflections, and stretches.

• Parent Function: or

• Transformation:

• Steps:

  1. Identify vertical and horizontal asymptotes.

  2. Apply shifts: (horizontal), (vertical).

  3. Reflect if is negative.

  4. Plot test points and sketch curves approaching asymptotes.

• Example: VA at , HA at .

Graphing Procedure

• Step-by-Step:

  1. Factor numerator and denominator.

  2. Find domain by setting denominator = 0.

  3. Find holes by setting canceled common factors = 0.

  4. Find x-intercepts by setting numerator = 0.

  5. Find y-intercept by evaluating .

  6. Find vertical and horizontal/slant asymptotes.

  7. Determine intervals between asymptotes and plot points in each.

  8. Connect points and draw curves approaching asymptotes.

• Example:

  • Domain:

  • VA:

  • HA: Degrees equal, so

  • x-intercept:

  • y-intercept:

Practice Example

• Graph

  • Factor denominator:

  • Domain:

  • VA: ,

  • HA: Degree numerator < denominator, so

  • x-intercept:

  • y-intercept:

Summary Table: Graphing Rational Functions

Key Formulas and Concepts

• General Rational Function:

• Domain:

• Vertical Asymptote: (after simplification)

• Horizontal Asymptote:

  • If :

  • If :

• Hole: Set canceled common factor

Additional info: Academic context and examples were expanded for clarity and completeness. Tables were reconstructed to summarize procedures and properties.