Vertical Asymptotes

Overview

• Vertical asymptotes are lines that occur when the denominator of a rational function equals zero, causing the function to become undefined.
• Unlike holes, vertical asymptotes occur when a factor of the denominator does not cancel out with a factor in the numerator.
• Represented by a dotted vertical line on graphs.

Identifying Vertical Asymptotes

1. Rational Functions: Typically in the form (\frac{P(x)}{Q(x)}).
2. Set denominator equal to zero: Solve the equation (Q(x) = 0) to find potential vertical asymptotes.
3. Check for common factors: Ensure factors are not canceled by the numerator to confirm genuine asymptotes.
4. Equation and Graph: Vertical asymptotes are denoted as lines (x = a), where (a) is the value making the denominator zero.

Examples

Example 1

• Function: (f(x) = \frac{2x + 3}{x^2 - 9})
• Vertical asymptotes at (x = 3) and (x = -3) after setting ((x+3)(x-3) = 0).
• Holes: If (x \pm 3) were in both numerator and denominator, making them cancel, they would represent holes instead.

Example 2

• Function Criteria:
  • Vertical asymptotes at 0 and 3
  • Zeroes at 2 and 5
  • Hole at (4, 2)
• Construct function: (f(x) = \frac{4(x-2)(x-5)(x-4)}{x(x-3)(x-4)})

Steps to Find Vertical Asymptotes

1. Identify Rational Function: (\frac{P(x)}{Q(x)}).
2. Denominator Zero: Solve (Q(x) = 0) for potential asymptotes.
3. Common Factors: Exclude factors canceled by the numerator.
4. List Asymptotes: Remaining (x) values are vertical asymptotes.

Practice and Application

• Identify asymptotes, holes, and domain restrictions for various rational functions by setting denominators to zero and analyzing factors.

Key Takeaways

• Vertical asymptotes indicate where functions become undefined.
• They differ from holes, which occur when factors cancel out.
• Essential for understanding function behavior and graphing rational expressions.