AP Pre-Calculus Unit 1 Review Lecture

Speakers: Jamal Sadiki and Becky Barnes

Introduction

• First AP Pre-Calculus exam
• Teachers: Jamal from East Bridgewater Junior Senior High School, Becky from West Springfield High School
• Focus: Unit 1 Review

Multiple Choice Question Analysis

• Polynomial Function P: ( P(x) = ax^n + b )
  • ( a, b ) are non-zero constants, ( n ) is a positive integer
  • As ( x \to \infty, P(x) \to -\infty )
  • Analyze the sign of ( a ) and behavior of ( P(x) )

Key Concepts

• Input Values: Independent variable behavior (( x \to \infty ))
• Output Values: Dependent variable behavior (( P(x) \to -\infty ))
• Limit Analysis:
  • Discard options where output goes to ( +\infty )

Polynomial End Behavior

• End Behavior Analysis:
  • Even degree, positive leading coefficient (e.g., ( x^4 )): Both ends ( \to \infty )
  • Even degree, negative leading coefficient: Both ends ( \to -\infty )
  • Odd degree, positive leading coefficient (e.g., ( x^3 )): Left ( \to -\infty ), Right ( \to \infty )
  • Odd degree, negative leading coefficient: Left ( \to \infty ), Right ( \to -\infty )

Dominant Terms in Polynomials

• Dominance Concept:
  • Higher powers of ( x ) dominate as ( x ) increases
  • Example: ( x^5 ) dominates lower degree terms as ( x \to \infty )

Rational Functions and End Behavior

• Rational Function Example:
  • ( f(x) = \frac{x^2 + 3x - 2}{x^3 - 5x^2 + 4x - 9} )
  • End behavior determined by dominant terms
  • Limits: ( \lim_{x \to -\infty} = 0 ), ( \lim_{x \to \infty} = 0 ) (Denominator dominates)

Horizontal Asymptotes

• Behavior for rational functions often results in horizontal asymptotes if degrees of top and bottom are the same or bottom is higher

Clarifying Coefficients

• Example: Different coefficients affect horizontal asymptote
  • ( f(x) = \frac{9x^2 - 2x - 1}{7x^3 + 3x^2 - 4x + 8} )
  • Same degree simplification to constant ( = \frac{-2}{7} )

Multiplicity and Vertical Asymptotes

• Multiplicity Definition: Number of factor repetitions for zeros
• Classification Based on Multiplicity:
  • Numerator's multiplicity ( \ge ) Denominator's: Hole
  • Numerator's multiplicity (<) Denominator's: Vertical asymptote

Practice Problem Example

• Problem: Factor and analyze given rational expressions
  • Determine vertical asymptotes and holes from zeros and multiplicity

Important Terms

• Increase/Decrease Without Bound: Indicates ( x \to \infty ) or ( x \to -\infty )
• End Behavior: Examine limits as ( x \to \pm\infty )
• Multiplicity: Impacts classification as hole or vertical asymptote

Conclusion

• Key focus on limit notation and asymptote behavior
• Preparation tips for AP Pre-Calculus exam

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Next Session: Focus on Unit 2.