Notes for Section 4.2: Graphing Rational Functions

Section 4.2 Part 1: Graphing Rational Functions

Essential Question

  • How can you graph a rational function?

Example 1: Rewrite a Rational Function to Identify Asymptotes

  • Given Function: g(x) = ( \frac{4x}{x-3} )
  • Related Function: f(x) = ( \frac{1}{x} )
  • Process: Use long division to express the rational function in a form that identifies asymptotes.
  • Rewrite g: Long division converts ( g(x) ) into the form ( a + \frac{k}{x-h} ).
  • Expression Derivation:
    • Long division steps lead to:
    • ( g(x) = 4 + \frac{12}{x-3} )
  • Vertical Asymptote:
    • The function has a vertical asymptote at ( x = 3 ) which is 3 units to the right of the vertical asymptote of f.
  • Horizontal Asymptote:
    • The horizontal asymptote of g is at ( y = 4 ), which is 4 units above the horizontal asymptote of f.
  • Graphing Details:
    • Sketch the graph based on the identified asymptotes and transformations.

Example 2: Rewrite Additional Rational Functions to Identify Asymptotes

  • Task: Rewrite rational function and identify asymptotes.
  • Function a: ( f(x) = \frac{6x}{x+1} )
    • Long Division Process:
    1. Divide 6x by x to yield 6.
    2. Remaining expression: ~values lead to the identification of horizontal and vertical asymptotes.
  • Function b: ( g(x) = \frac{x - 6}{x} )
    • Identify Asymptotes using Structure

Concept of Rational Functions

  • A Rational Function is defined as:
    • ( R(x) = \frac{P(x)}{Q(x)} )
    • Where both ( P(x) ) and ( Q(x) ) are polynomials.
  • Domain of Rational Function: All values of x for which ( Q(x) \neq 0 ).
    • Common issue related to exclusions from the domain due to vertical asymptotes.

Example 3: Finding Multiple Vertical Asymptotes of a Rational Function

  • Function: ( f(x) = \frac{3x - 2}{(x + 3)(x + 4)} )
  • Steps to find Vertical Asymptotes: Set the denominator equal to zero:
    • ( (x + 3)(x + 4) = 0 )
    • Solutions yield vertical asymptotes at ( x = -3 ) and ( x = -4 ).

Section 4.2 Part 2: Types of Horizontal Asymptotes

  • Horizontal asymptotes describe the end behavior of the function as ( |x| ) approaches infinity.
Case 1: Degree of Numerator Less than Degree of Denominator
  • Condition: When the degree of the numerator ( < ) degree of the denominator
  • Asymptote: There is a horizontal asymptote at ( y = 0 ).
  • Example: For a function ( g(x) ), as ( |x| \to \infty ), the function approaches 0.
Case 2: Degree of Numerator Greater than Degree of Denominator
  • Condition: Degree of the numerator ( > ) degree of the denominator
  • Asymptote: There is no horizontal asymptote; the function approaches ±∞ as x approaches infinity.
Case 3: Degree of the Numerator Equal to Degree of the Denominator
  • Condition: Degree of the numerator = degree of the denominator
  • Asymptote: The horizontal asymptote is at ( y = \frac{a}{b} ), where a and b are leading coefficients of the numerator and denominator respectively.

Example 6: Graphing a Function of the Form ( ax + b )

  • Function: ( f(x) = 2x + 1 )
  • Find Intercepts:
    • Vertical intercept: Set ( x = 0 ) to find ( y ): ( (0, 1) )
    • Horizontal intercept: Set ( y = 0 ): Results in ( x = -\frac{1}{2} )

Additional Conceptual Understanding

  • Real-World Application:
    • Example using time-distance logic to illustrate reciprocal functions where distance is fixed, and rate changes represent the horizontal asymptote.

Try It! Exercise

  • Task: Find horizontal asymptotes for given functions.
  • Function a: ( h(x) = 2x^2 + x - 9 ) - no horizontal asymptote.
  • Function b: ( k(x) = \frac{2x - 8}{x^2 - 12} ) - horizontal asymptote at ( y = 0 ) based on the analysis of degrees of numerator and denominator.