Graphing Rational Functions and Their Asymptotes

Introduction

• Rational functions: expressed as quotients of polynomial functions.
• Graphing rational functions involves complexities due to:
  • Behavior unpredictability.
  • Domain limitations (denominator cannot be zero).

Example: Graphing ( \frac{1}{x} )

• Domain: Excludes ( x = 0 ).
• Behavior: As ( x ) approaches zero or infinity.
  • As ( x \to -\infty ): ( f(x) \to 0 ).
  • As ( x \to 0^- ): ( f(x) \to -\infty ).
  • As ( x \to \infty ): ( f(x) \to 0 ).
  • As ( x \to 0^+ ): ( f(x) \to \infty ).

Asymptotes

• Types of Asymptotes:
  • Vertical: Lines ( x = a ) where the function is undefined.
  • Horizontal: ( y = b ) where function approaches as ( x \to \infty ).
  • Possible to have oblique asymptotes (not covered in detail).

Finding Asymptotes

• Vertical Asymptotes:
  • Occur at zeros of the denominator.
  • Example: ( \frac{x}{x^2-9} ) has vertical asymptotes at ( x = -3, 3 ).
• Horizontal Asymptotes:
  • Determine by comparing degrees of numerator and denominator.
  • If degree of denominator > numerator: asymptote at ( y = 0 ).
  • If degrees are equal: ( y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} ).
  • If degree of numerator > denominator: no horizontal asymptote.

Graphing Strategy

1. Vertical Asymptotes: Find zeros of the denominator.
2. Horizontal Asymptotes: Compare polynomial degrees.
3. X and Y Intercepts: Identify for sketching.
4. Select Points to Plot: Understand behavior around asymptotes.

Transformations

• Apply transformation rules similar to polynomials.
• Vertical shifts, horizontal shifts, and stretches/compressions.

Conclusion

• Utilize asymptote identification and point plotting for graphing.
• Rational functions can be manipulated using transformation techniques.
• Practice graph comprehension with given concepts.