(L16) Graphing Functions in Business Calculus: Guidelines, Analysis, and Examples

Graphing Functions: Step-by-Step Guidelines

Overview

Graphing a function is a fundamental skill in Business Calculus, allowing students to visualize and analyze the behavior of mathematical models. The following guidelines outline a systematic approach to sketching the graph of a function f(x), including domain analysis, intercepts, asymptotes, symmetry, intervals of increase/decrease, extrema, concavity, and inflection points.

1. Determining the Domain

• Domain: The set of all possible input values (x) for which f(x) is defined.

• Exclude values that cause division by zero, negative values under even roots, or negative arguments for logarithms.

• Example: For f(x) = x^4 - 4x^3, the domain is $(-, )$.

2. Finding Intercepts

• Y-intercept: Set x = 0 and solve for f(0).

• X-intercepts: Solve f(x) = 0 for x.

• Example: For f(x) = x^4 - 4x^3, f(0) = 0 (y-intercept), and x-intercepts are found by solving $x^4 - 4x^3 = 0$.

3. Asymptotes

• Vertical Asymptotes: Occur where the function is undefined due to division by zero.

• Horizontal Asymptotes: Determined by evaluating limits as x approaches infinity.

• Example: For f(x) = \frac{x}{x^2 + 1}, as x \to \pm\infty, $\lim_{x \to \pm\infty} \frac{x}{x^2 + 1} = 0$ (horizontal asymptote at y = 0).

4. Symmetry

• Even Function: $f(-x) = f(x)$; graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$; graph is symmetric about the origin (180° rotation).

• Example: f(x) = \frac{x}{x^2 + 1} is odd.

5. Intervals of Increase/Decrease

• Find where the first derivative $f'(x)$ is positive (increasing) or negative (decreasing).

• Critical Points: Where $f'(x) = 0$ or $f'(x)$ does not exist.

• Example: For f(x) = x^4 - 4x^3, $f'(x) = 4x^3 - 12x^2$; solve $f'(x) = 0$ for critical points.

6. Relative Extrema

• Local Maximum: $f'(x)$ changes from positive to negative.

• Local Minimum: $f'(x)$ changes from negative to positive.

• Use the First Derivative Test to classify extrema.

• Example: For f(x) = x^4 - 4x^3, local minimum at $x = 3$.

7. Concavity and Inflection Points

• Concave Up: $f''(x) > 0$

• Concave Down: $f''(x) < 0$

• Inflection Point: Where $f''(x)$ changes sign.

• Example: For f(x) = x^4 - 4x^3, $f''(x) = 12x^2 - 24x$; inflection points at $x = 0$ and $x = 2$.

8. Sketching the Graph

• Plot intercepts, asymptotes, critical points, and inflection points.

• Indicate intervals of increase/decrease and concavity.

• Connect points smoothly, respecting the behavior indicated by derivatives.

Table: Shape of the Graph Based on Derivatives

The following table summarizes the general shape of the graph depending on the sign of the first and second derivatives:

	f' is increasing	f' is decreasing
f is concave up (+)	Upward curve, rising	Upward curve, falling
f is concave down (-)	Downward curve, rising	Downward curve, falling

Worked Example: Sketching f(x) = x^4 - 4x^3

Step-by-Step Analysis

• Domain: $(-, )$

• Intercepts: $f(0) = 0$; x-intercepts at $x = 0$ and $x = 4$

• Asymptotes: None

• Symmetry: Not even, not odd

• Critical Points: $f'(x) = 4x^3 - 12x^2 = 4x^2(x - 3)$; critical points at $x = 0, x = 3$

• Intervals of Increase/Decrease:

  • Increasing on $(3, )$

  • Decreasing on $(-, 0)$ and $(0, 3)$

• Local Minimum: At $x = 3$, $f(3) = 3^4 - 4 imes 3^3 = 81 - 108 = -27$

• Inflection Points: $f''(x) = 12x^2 - 24x = 12x(x - 2)$; inflection points at $x = 0$ and $x = 2$

Worked Example: Sketching f(x) = x/(x^2 + 1)

Step-by-Step Analysis

• Domain: $(-, )$

• Intercepts: $f(0) = 0$; x-intercept at $x = 0$

• Asymptotes: Horizontal asymptote at $y = 0$

• Symmetry: Odd function

• Critical Points: $f'(x) = \frac{1 - x^2}{(x^2 + 1)^2}$; critical points at $x = 1, x = -1$

• Intervals of Increase/Decrease:

  • Increasing on $(-1, 1)$

  • Decreasing on $(-, -1)$ and $(1, )$

• Inflection Points: $f''(x) = \frac{2x(x^2 - 3)}{(x^2 + 1)^3}$; inflection points at $x = 0, x = , x = -$

Summary of Key Concepts

• Domain and Range are foundational for understanding where a function is defined and its possible outputs.

• Intercepts provide anchor points for graphing.

• Asymptotes indicate long-term behavior.

• Symmetry simplifies graphing and analysis.

• Critical Points and Extrema reveal local maxima and minima.

• Concavity and Inflection Points describe the curvature and changes in the graph's shape.

Additional info:

• These notes expand on the original handwritten and typed content, providing full academic context and step-by-step analysis for Business Calculus students.

• All examples and tables are reconstructed and explained for clarity and completeness.