The curve sketching checklist
A sketch is not an exact plot. It shows the right shape and the important points. Work through this list:
- Domain: which x-values are allowed? (No division by 0, no square root of a negative.)
- Intercepts: put x = 0 to get the y-intercept; solve f(x) = 0 for x-intercepts.
- Symmetry: if f(−x) = f(x) the graph is symmetric about the y-axis (even); if f(−x) = −f(x) it is symmetric about the origin (odd).
- Stationary points: solve f'(x) = 0.
- Sign table (variation table): where f'(x) > 0 the curve rises; where f'(x) < 0 it falls.
- Concavity: f''(x) > 0 means cup-shaped (∪); f''(x) < 0 means cap-shaped (∩).
- Asymptotes and ends: what happens as x → ±∞ and near points not in the domain?
- Join the points with a smooth curve that obeys all of the above.
Stationary points: maximum, minimum, or neither
A stationary point is where the tangent is flat: f'(x) = 0. Decide its type in one of two ways.
First-derivative test: look at the sign of f' just before and after. + then − gives a local maximum; − then + gives a local minimum; no change of sign gives a stationary point of inflection (e.g. y = x³ at 0).
Second-derivative test: f''(x) < 0 → maximum; f''(x) > 0 → minimum; f''(x) = 0 → test fails, use the sign table.
The global (absolute) maximum on an interval [a, b] is the biggest value among the local maxima and the end values f(a), f(b).
Concavity and points of inflection
The second derivative tells how the slope changes. If f''(x) > 0 the slope keeps increasing and the curve holds water like a cup (concave up). If f''(x) < 0 it spills water like a cap (concave down).
A point of inflection is where the concavity changes. Find where f''(x) = 0 and check that f'' really changes sign there. For y = x⁴, f''(0) = 0 but f'' does not change sign, so (0, 0) is a minimum, not an inflection.
Asymptotes and rational functions
An asymptote is a line the curve gets closer and closer to.
- Vertical: where the denominator is 0 and the numerator is not. For y = (x + 1)/(x − 2), x = 2.
- Horizontal: the limit of y as x → ±∞. For y = (x + 1)/(x − 2), y → 1.
- Oblique: when the top degree is one more than the bottom, divide: (x² + 1)/x = x + 1/x, so y = x is an asymptote.
On a sketch draw asymptotes as dashed lines first, then fit the curve around them.
Graphs in economics and the length of a curve
Economics: cost C(q), revenue R(q) and profit P(q) = R(q) − C(q) are functions of quantity q. Profit is greatest where P'(q) = 0 and P''(q) < 0, which is where marginal revenue equals marginal cost (R'(q) = C'(q)). Average cost C(q)/q is often U-shaped.
Arc length: the length of a smooth curve y = f(x) from x = a to x = b is
L = ∫ab √(1 + (f'(x))²) dx.
It adds up tiny slanted pieces, each of length √(dx² + dy²). For motion, the distance travelled from t = a to t = b is ∫ |v(t)| dt.
Key formulas and definitions
- Stationary points: f'(x) = 0
- f'(x) > 0 → increasing; f'(x) < 0 → decreasing
- f''(x) < 0 at a stationary point → maximum; f''(x) > 0 → minimum
- Inflection: f''(x) = 0 AND f'' changes sign
- Even: f(−x) = f(x); odd: f(−x) = −f(x)
- Arc length: L = ∫ₐᵇ √(1 + (f'(x))²) dx
Worked examples
1. Find the stationary points of y = x² − 4x + 1 and their type.
y' = 2x − 4 = 0 → x = 2, y = 4 − 8 + 1 = −3. y'' = 2 > 0, so (2, −3) is a minimum. The parabola opens upward.
2. Sketch y = x³ − 3x.
Intercepts: (0, 0), (±√3, 0). y' = 3x² − 3 = 0 → x = ±1: (−1, 2) and (1, −2). Signs of y': + − +, so max at (−1, 2), min at (1, −2). y'' = 6x: inflection at (0, 0). Ends: up on the right, down on the left. Odd function, symmetric about the origin.
3. Sketch y = x³ − 6x² + 9x.
y = x(x − 3)², so intercepts (0, 0) and (3, 0), touching at x = 3. y' = 3x² − 12x + 9 = 3(x − 1)(x − 3) = 0 → x = 1, 3. y(1) = 4, y(3) = 0. y'' = 6x − 12: y''(1) = −6 → max (1, 4); y''(3) = 6 → min (3, 0). Inflection where 6x − 12 = 0: x = 2, y = 2.
4. Find the asymptotes of y = (2x + 1)/(x − 1) and the intercepts.
Vertical: x = 1. Horizontal: y → 2 as x → ±∞. y-intercept: x = 0 → y = −1. x-intercept: 2x + 1 = 0 → x = −0.5. y' = −3/(x − 1)² < 0, so the curve falls on each side of x = 1.
5. Profit P(q) = −2q² + 40q − 50 (thousand ₹ or €). Find the best quantity and the largest profit.
P'(q) = −4q + 40 = 0 → q = 10. P''(q) = −4 < 0, so it is a maximum. P(10) = −200 + 400 − 50 = 150 thousand.
6. Find the arc length of y = (2/3)x^(3/2) from x = 0 to x = 3.
y' = x^(1/2), so 1 + (y')² = 1 + x. L = ∫₀³ √(1 + x) dx = (2/3)(1 + x)^(3/2) from 0 to 3 = (2/3)(8 − 1) = 14/3 ≈ 4.67 units.
Common mistakes
- Saying every point with f'(x) = 0 is a max or min. y = x³ has f'(0) = 0 but keeps rising: it is an inflection.
- Calling every point with f''(x) = 0 an inflection. You must check that f'' changes sign (y = x⁴ at 0 does not).
- Forgetting the end values when finding the greatest or least value on a closed interval.
- Drawing the curve crossing a vertical asymptote. The function is not defined there.