Algebra 3.5 notes

Chapter 3: Polynomial and Rational Functions

• Covering key aspects of Rational Functions and their Graphs.

Section 3.5: Rational Functions and Their Graphs

• Topics to cover:

  • Finding domains of rational functions.

  • Understanding arrow notation.

  • Identifying vertical and horizontal asymptotes.

  • Graphing rational functions using transformations.

  • Identifying slant asymptotes.

  • Solving applied problems with rational functions.

Rational Functions

• Definition: Quotients of polynomial functions, expressed as ( \frac{p(x)}{q(x)} ) where ( p ) and ( q ) are polynomial functions.

• Domain: Set of all real numbers except for x-values that make the denominator zero.

Finding the Domain of a Rational Function

• Example 1a: Domain of a function is found by excluding x-values causing the denominator to be zero.

  • Domain: All real numbers except for specified values (e.g., ( 5 )).

• Example 1b: Same approach for another rational function, with domain adjustments as necessary.

• Example 1c: If no real numbers cause the denominator to equal zero, the domain includes all real numbers.

Arrow Notation

• Describes behavior of functions as they approach particular values.

Vertical Asymptotes

• Definition (1 of 2): ( x = a ) is a vertical asymptote if the function increases without bound as ( x ) approaches ( a ).

• Definition (2 of 2): ( x = a ) is a vertical asymptote if the function decreases without bound as ( x ) approaches ( a ).

• Locating Vertical Asymptotes:

  • If ( f ) is a rational function without common factors and ( a ) is a zero of the denominator, then ( x = a ) is a vertical asymptote.

Examples of Vertical Asymptotes

• Example 2a: For a function with zeros in the denominator:

  • Found asymptotes at lines corresponding to zeros:

    • Line 1 and line 2 as vertical asymptotes.

• Example 2c: If unable to factor the denominator and have no real zeros, the function has no vertical asymptotes.

Horizontal Asymptotes

• Definition: Horizontal line ( y = b ) is a horizontal asymptote of ( f ) if ( f ) approaches ( b ) as ( x ) approaches ±∞.

• Locating Horizontal Asymptotes:

  • Relation between degrees of numerator (n) and denominator (m):

    • If ( n < m ), asymptote is the x-axis ( y = 0 ).

    • If ( n = m ), asymptote is ( y = \frac{C_1}{C_2} ) (leading coefficients of numerator and denominator).

    • If ( n > m ), no horizontal asymptote.

Examples of Horizontal Asymptotes

• Example 3a: Given the degrees are equal, found horizontal asymptote using leading coefficients.

• Example 3b: Degree analysis leads to identifying the x-axis as a horizontal asymptote.

• Example 3c: Degree of the numerator greater than the denominator means no horizontal asymptote.

Basic Rational Functions

• Odd Function: Exhibits origin symmetry.

• Even Function: Exhibits y-axis symmetry.

Using Transformations to Graph a Rational Function

• Steps outlined to shift graph based on transformations for rational functions.

• Asymptotes must be re-evaluated post-transformation.

Strategy for Graphing Rational Functions

• Determine graph symmetry.

• Find y-intercept by evaluating ( f(0) ).

• Find x-intercepts and vertical asymptotes.

• Determine horizontal asymptotes.

• Plot points for accurate graph representation.

Slant Asymptotes

• Definition: Present when degree of numerator is one more than degree of denominator.

• How to find: Divide numerator by denominator to form the equation of the slant asymptote.

Applications

• Example 9a: Cost function from fixed costs and variable production costs.

• Example 9b: Average cost function derived from total costs.

• Example 9c: Calculate average costs at various production levels and interpret results.

• Identify horizontal asymptote which describes limiting behavior as production increases.