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{{x \to a^+}} f(x) = \infty or \lim{{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{{x \to \infty}} f(x) = b or \lim{{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