Curve Sketching

Curve Sketching

4.3 Vertical Asymptotes

• Definition: The line x = a is a vertical asymptote of the graph of a function f if:

  • $\lim<em>{{x \to a^+}} f(x) = \infty$ or $\lim</em>{{x \to a^-}} f(x) = \infty$

• Note: Although a vertical asymptote is not part of the graph, it is a useful aid for sketching the graph.

Finding Vertical Asymptotes of Rational Functions

• Suppose f is a rational function defined as:

  • $f(x) = \frac{P(x)}{Q(x)}$

  • where P and Q are polynomial functions.

• Condition for vertical asymptote:

  • If $Q(a) = 0$ but $P(a) \neq 0$, then the line x = a is a "two-sided" vertical asymptote of the graph of f.

Horizontal Asymptotes

• Definition: The line y = b is a horizontal asymptote of the graph of a function f if:

  • $\lim<em>{{x \to \infty}} f(x) = b$ or $\lim</em>{{x \to -\infty}} f(x) = b$

Example 1: Finding Asymptotes for a Rational Function

• Function: $f(x) = \frac{2x^2}{x^2 - 16}$

• Vertical Asymptotes: Identify where the denominator equals zero:

  • $x^2 - 16 = 0$

  • Solutions: $x = 4$ and $x = -4$

• Horizontal Asymptotes: Check limits as x approaches ±∞:

  • $\lim_{{x \to \infty}} f(x) = 2$

  • $\lim_{{x \to -\infty}} f(x) = 2$

• Conclusion: Vertical asymptotes at x = 4 and x = -4, horizontal asymptote at y = 2.

Curve Sketching Guide

1. Determine: The domain of f.

2. Find: The x-intercepts and y-intercepts of f.

3. Find: All asymptotes of the graph of f.

4. Determine Behavior:

   • Evaluate the limits for large absolute values of x by calculating:

     • $\lim_{{x \to -\infty}} f(x)$

     • $\lim_{{x \to \infty}} f(x)$

   • Identify any asymptotes.

5. Determine: The intervals where f is increasing and where f is decreasing.

6. Find: The critical points (local extrema) of f by solving $f'(x) = 0$.

7. Determine: The concavity of the graph of f by analyzing $f''(x)$.

8. Find: The inflection points of f using $f''(x) = 0$.

9. Plot Additional Points: To further identify the shape of the graph and assist in sketching it.

Behavior of Graph based on First and Second Derivatives

• Summary Table:

  • The interactions of the signs of $f'(x)$ and $f''(x)$ will provide implications on:

  • Increasing or decreasing behavior.

  • Concavity and inflection points.

Example 2: Sketch the Graph

• Function: $y = 8 + \frac{8x}{x^2}$

• Derivatives:

  • First Derivative: $y' = -\frac{8x + 16}{x^3}$

  • Second Derivative: $y'' = \frac{16x + 48}{x^4}$

• Note: The algebra of simplifying these derivatives follows similar steps as previous examples.

Supplemental Examples

Example 3: Asymptotes

• Function: $f(x) = \frac{3x^2 - 3}{x^2 - x - 2}$

• Tasks involve finding both vertical and horizontal asymptotes.

Example 4: Sketching the Graph

• Function: $f(x) = x^4 + 4 - 3x^3$

• Detailed steps for deriving the sketch are necessary, focusing on asymptotes, intercepts, and derivative analysis.

Extra Space for Note-taking

• Additional notes and personal revisions can be made in this section.

MATH 221 University of Delaware