Curve Sketching

Introduction

Curve sketching: graphical representation of functions using calculus tools. Purpose: understand shape, behavior, key features of graphs. Techniques: derivatives for slopes, concavity; limits for asymptotes and end behavior. Applications: optimization, physics, engineering, data visualization.

"Graphing functions by hand builds intuition about their behavior, bridging algebraic expressions and geometric interpretation." -- George B. Thomas Jr.

Function and Domain

Definition

Function: mapping input x to output y=f(x). Domain: set of allowable x-values. Domain restrictions impact graph shape and existence.

Determining Domain

Analyze denominators, radicands, logarithms for restrictions. Example: f(x) = 1/(x-2) domain excludes x=2.

Impact on Graph

Discontinuities, gaps, vertical asymptotes arise where function undefined. Domain limits graph extent.

Intercepts

x-Intercepts

Points where graph crosses x-axis: solve f(x)=0. Number and multiplicity affect graph shape.

y-Intercepts

Point where graph crosses y-axis: evaluate f(0) if 0 in domain.

Role in Sketching

Anchors graph, provides initial points to connect with behavior analysis.

Symmetry

Even Functions

Condition: f(-x) = f(x). Graph symmetric about y-axis.

Odd Functions

Condition: f(-x) = -f(x). Graph symmetric about origin.

Periodic and Other Symmetries

Periodicity: f(x + p) = f(x). Symmetry simplifies graphing and reduces domain needed.

Critical Points and Extrema

Definition

Critical points: x-values where f'(x)=0 or f'(x) undefined. Potential maxima, minima, or saddle points.

Finding Critical Points

Calculate first derivative, solve f'(x)=0, check domain.

Role in Graph Shape

Indicate peaks, valleys, horizontal tangents. Essential for shape analysis.

First Derivative Test

Concept

Analyze sign changes of f'(x) around critical points to classify extrema.

Procedure

Evaluate f'(x) left and right of critical point:

• Positive to negative: local max
• Negative to positive: local min
• No sign change: neither

Limitations

Indeterminate if f'(x) does not change sign; use second derivative or other tests.

Second Derivative Test

Concept

Uses concavity to classify critical points. Requires f''(x) at critical point.

Procedure

If f''(c) > 0, local minimum at c.
If f''(c) < 0, local maximum at c.
If f''(c) = 0, test inconclusive.

Advantages and Drawbacks

Simpler than first derivative test when applicable; fails for inflection points.

Concavity and Inflection Points

Concavity

Determined by sign of second derivative f''(x):

• f''(x) > 0: graph concave up (cup-shaped)
• f''(x) < 0: graph concave down (cap-shaped)

Inflection Points

Points where concavity changes sign; f''(x) = 0 or undefined and sign changes around point.

Significance

Indicate shape transitions; important for accurate graph curvature.

Asymptotes

Vertical Asymptotes

Lines x = a where f(x) → ±∞ as x → a. Typically where denominator zero or discontinuities.

Horizontal Asymptotes

Lines y = L where f(x) → L as x → ±∞. Indicate end behavior.

Oblique (Slant) Asymptotes

Lines y = mx + b approached by f(x) as x → ±∞ when no horizontal asymptote exists.

Finding Vertical Asymptotes:1. Find values where denominator = 0.2. Verify limits ±∞ at those points.Finding Horizontal Asymptotes:1. Evaluate limits as x → ±∞.2. If limit = L finite, y = L asymptote.Finding Oblique Asymptotes:1. Perform polynomial division if degree numerator = degree denominator + 1.2. Use quotient as asymptote equation.

Behavior at Infinity

End Behavior Analysis

Analyze limits as x → ±∞ to understand graph tails.

Dominant Terms

Highest degree terms dictate growth/decay rates.

Classification

Polynomial: heads to ±∞ or finite limits.
Rational: horizontal or oblique asymptotes.
Exponential/logarithmic: rapid growth/decay.

Sketching Strategy

Step 1: Determine Domain

Identify restrictions and discontinuities.

Step 2: Find Intercepts and Symmetry

Plot key points, reduce graphing domain if symmetric.

Step 3: Compute Derivatives

Locate critical points, inflection points.

Step 4: Analyze Derivative Signs

Determine increasing/decreasing intervals, concavity.

Step 5: Identify Asymptotes

Vertical, horizontal, or oblique.

Step 6: Combine Information

Plot points and behavior, sketch smooth curve.

Examples

Example 1: Polynomial Function

f(x) = x³ - 3x² + 2

Domain: all real numbers. Derivatives:

f'(x) = 3x² - 6xf''(x) = 6x - 6

Critical points: f'(x) = 0 → x=0, x=2

Second derivative test:

• f''(0) = -6 < 0 → local max at x=0
• f''(2) = 6 > 0 → local min at x=2

Inflection point where f''(x)=0 → x=1

Graph: max at (0,2), min at (2,-2), inflection at (1,0).

Example 2: Rational Function

f(x) = (x² - 1) / (x - 2)

Domain: x ≠ 2 (vertical asymptote)

Horizontal or oblique asymptote by division:

Divide numerator by denominator:(x² - 1) ÷ (x - 2) = x + 2 + remainderAs x → ±∞, f(x) ~ x + 2 (oblique asymptote)

Critical points found via first derivative; analyze increasing/decreasing behavior accordingly.

References

• Stewart, J. "Calculus: Early Transcendentals," Brooks Cole, 8th ed., 2015, pp. 120-165.
• Thomas, G. B., Weir, M. D., Hass, J. "Thomas' Calculus," Pearson, 14th ed., 2017, pp. 200-240.
• Anton, H., Bivens, I., Davis, S. "Calculus," Wiley, 10th ed., 2012, pp. 150-190.
• Edwards, C. H., Penney, D. E. "Calculus and Analytic Geometry," Pearson, 7th ed., 2002, pp. 300-350.
• Purcell, E. J. "Calculus with Analytic Geometry," McGraw-Hill, 1998, pp. 180-220.