Summary of Curve Sketching Techniques

Essential Concepts for Curve Sketching

• Analyzing and sketching the graphs of functions requires identifying key characteristics to ensure accurate representation.
• Essential graphical components include:
  • $x$-intercepts and $y$-intercepts
  • Domain and range
  • Continuity and differentiability
  • Relative extrema (maxima and minima)
  • Concavity and points of inflection
  • Vertical asymptotes and horizontal asymptotes

Guidelines for Analyzing Graphs

1. Determine the domain and range of the function, accounting for the context if the function models a real-life situation.
2. Locate all intercepts and determine the asymptotes ($x = c$ or $y = L$).
3. Find the values of $x$ where $f'(x) = 0$ or where the derivative is undefined to identify critical numbers and relative extrema.
4. Find the values of $x$ where $f''(x) = 0$ or where the second derivative is undefined to identify points of inflection and determine concavity.

Properties of Polynomial Functions

• A polynomial function of degree $n$ has a maximum of $n - 1$ relative extrema and at most $n - 2$ points of inflection.
• Polynomial functions of even degree must possess at least one relative extremum.
• Low-degree polynomial functions (degrees $0, 1, 2, \text{ and } 3$) are frequently used as mathematical models due to their simplicity.

Analysis of Specific Function Examples

Example 2: Rational Polynomial

• Function: $f(x) = x^4 - 12x^3 + 48x^2 - 64x$
• Intercepts: $(0, 0)$ and $(4, 0)$.
• Extrema: Relatve minimum at $(1, -27)$.
• Points of Inflection: Occur at $(2, -16)$ and $(4, 0)$.
• Analysis of Figure 3.57 confirms the critical numbers are $x = 1$ and $x = 4$.

Example 4: Rational Function with Asymptotes

• Function: $f(x) = \frac{2(x^2 - 9)}{x^2 - 4}$
• Domain: All real numbers except $x = \text{-}2$ and $x = 2$.
• Intercepts: $x$-intercepts at $(\text{-}3, 0)$ and $(3, 0)$.
• Asymptotes: Vertical asymptotes at $x = \text{-}2$ and $x = 2$; horizontal asymptote at $y = 2$.
• Extrema: Relative minimum at $(0, 4.5)$.
• Inflection: No points of inflection, as verified by Figure 3.59.