Vertical Asymptotes - Algebra-2

1. Fundamental Concepts

Definition: Vertical asymptotes are vertical lines that the graph of a function approaches but never touches. They occur where the denominator of a rational function is zero and the numerator is not zero at that point.
Identifying Asymptotes: To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
Behavior Near Asymptotes: As $x$ approaches the value that makes the denominator zero, the function values can either increase or decrease without bound.

Basic Rule: $$f(x) = \frac{p(x)}{q(x)}, \text{ where } q(x) \neq 0$$

Degree Preservation: Vertical asymptotes occur where $q(x) = 0$ and $p(x) \neq 0$

Application: Used to analyze the behavior of rational functions and their graphs

Problem: Find the vertical asymptotes of $f(x) = \frac{x + 2}{x - 3}$.

Validation: The function $f(x)$ approaches infinity as $x$ approaches 3 from either side.

Problem: Find the vertical asymptotes of $g(x) = \frac{x^2 - 4}{x^2 - 9}$.

Factor both the numerator and the denominator: $g(x) = \frac{(x + 2)(x - 2)}{(x + 3)(x - 3)}$
Set the denominator equal to zero: $(x + 3)(x - 3) = 0$
Solve for $x$: $x = -3$ and $x = 3$

Validation: The function $g(x)$ approaches infinity as $x$ approaches $-3$ and $3$ from either side.

Factorization Strategy: Always factor the numerator and the denominator to identify common factors and potential holes in the graph.
Error-Proofing: Double-check by substituting values close to the potential asymptote points into the original function.
Concept Reinforcement: Understand the relationship between the zeros of the denominator and the vertical asymptotes.