Lesson 3: Curve Sketching
Estimated time: 40-50 minutes
Learning Objectives
- Follow a systematic curve-sketching procedure
- Combine information from f, f', and f'' to sketch accurate graphs
- Identify all key features: intercepts, asymptotes, extrema, inflection points
Curve Sketching Checklist
Systematic Curve Sketching Procedure
- Domain: Where is f defined?
- Intercepts: x-intercepts (set f = 0) and y-intercept (evaluate f(0)).
- Symmetry: Is f even, odd, or periodic?
- Asymptotes: Vertical (denominator = 0), horizontal (limits at ±∞).
- First derivative: Increasing/decreasing intervals and local extrema.
- Second derivative: Concavity and inflection points.
- Plot key points and connect the dots using all information above.
Worked Example
Example 1: Sketch f(x) = x³ − 3x²
Domain: All real numbers.
Intercepts: f(0) = 0. f(x) = x²(x − 3) = 0 at x = 0, 3.
Symmetry: Neither even nor odd.
Asymptotes: None (polynomial).
f'(x) = 3x² − 6x = 3x(x − 2). Critical points: x = 0, 2.
f' > 0 on (−∞, 0), f' < 0 on (0, 2), f' > 0 on (2, ∞).
Local max at (0, 0), local min at (2, −4).
f''(x) = 6x − 6. f'' = 0 at x = 1. Concave down on (−∞, 1), concave up on (1, ∞). Inflection point at (1, −2).
Sketch: Rises to (0, 0), falls to (2, −4), then rises through (3, 0) and beyond. Changes concavity at (1, −2).
Example 2: Sketch f(x) = x/(x² + 1)
Domain: All reals. Intercepts: (0, 0). Symmetry: f(−x) = −f(x), so f is odd.
Asymptotes: No vertical. Horizontal: limx→±∞ x/(x²+1) = 0, so y = 0.
f'(x) = (1 − x²)/(x² + 1)². f' = 0 at x = ±1.
f' > 0 on (−1, 1): increasing. f' < 0 outside: decreasing.
Local min at (−1, −1/2), local max at (1, 1/2).
f''(x) = 2x(x² − 3)/(x² + 1)³. Inflection at x = 0, ±√3.
Key Takeaways
- Follow the checklist systematically for accurate sketches.
- f' gives shape (rising/falling); f'' gives curvature (concave up/down).
- Plot the key points: intercepts, extrema, inflection points, and note asymptotic behavior.
- Check symmetry early; it can halve your work.
Check Your Understanding
1. What are the key features to plot for f(x) = x4 − 4x²?
2. Name the first step you should always take when sketching a curve.
3. If a function has a horizontal asymptote y = 3, what does that tell you about the graph?
4. For f(x) = x³ − 3x², identify the inflection point.