The Full Curve Sketching Process
Derivatives give you complete information about a function's behaviour: where it rises, where it falls, where it bends, and where it has extreme values. Systematic curve sketching turns this information into an accurate graph.
The 7-Step Checklist
📋 Systematic Method
Step 1Domain: where is f(x) defined?
Step 2Intercepts: find f(0) and solve f(x) = 0
Step 3Asymptotes: check x→±∞ and any discontinuities
Step 4Find f'(x): critical points where f'=0 or f' undefined
Step 5Sign chart for f': intervals of increase/decrease
Step 6Find f''(x): inflection points, concavity
Step 7Sketch: plot key points, connect with correct shape
First Derivative Test
At a critical point c: if f' changes from positive to negative, it is a local maximum. If f' changes from negative to positive, it is a local minimum. If f' does not change sign, it is neither.
Second Derivative Test
At a critical point c where f'(c) = 0: if f''(c) > 0, local minimum (concave up). If f''(c) < 0, local maximum (concave down). If f''(c) = 0, the test is inconclusive.
📋 Sketch f(x) = x³ − 3x
f'(x)= 3x²−3 = 3(x−1)(x+1). Critical points: x = ±1
Signf' > 0 on (−∞,−1) and (1,∞); f' < 0 on (−1,1)
ExtremaLocal max at x=−1: f(−1)=2. Local min at x=1: f(1)=−2
f''(x)= 6x. Inflection at x=0. Concave down x<0, up x>0
Frequently Asked Questions
What is a sign chart?▾
A sign chart tests the sign of f'(x) in each interval between critical points. Pick a test value in each interval, plug into f', and record + or −. Positive means increasing; negative means decreasing.
When does the second derivative test fail?▾
When f''(c) = 0. This happens at inflection points and flat extrema like x⁴ at x=0. Use the first derivative test instead — it always works.
What Derivatives Tell You About Shape
Before derivatives, you could only plot functions by calculating dozens of individual points. Derivatives give you structural information — you know exactly where the function rises, falls, bends up, bends down, and has peaks and valleys. This lets you sketch an accurate graph from far fewer calculations.
The first derivative f'(x) tells you slope: positive means rising, negative means falling, zero means a potential peak or valley. The second derivative f''(x) tells you curvature: positive means concave up (smiling), negative means concave down (frowning).
Complete Worked Example — f(x) = x³ − 6x² + 9x + 1
📋 Full curve sketch
DomainAll real numbers (polynomial)
y-interceptf(0) = 1. Point: (0, 1)
f'(x)= 3x² − 12x + 9 = 3(x−1)(x−3). Critical points: x = 1, x = 3
Sign of f'x<1: + (rising), 13: + (rising)
ExtremaLocal max at x=1: f(1)=5. Local min at x=3: f(3)=1
f''(x)= 6x − 12. f''=0 at x=2. f''<0 for x<2 (concave down), f''>0 for x>2 (concave up)
InflectionAt x=2: f(2) = 8−24+18+1 = 3. Inflection point: (2, 3)
End behaviourx→+∞: f→+∞. x→−∞: f→−∞ (odd-degree polynomial)
Curve Sketching Rational Functions
Rational functions require extra steps: find vertical asymptotes (where denominator = 0), horizontal asymptotes (compare degrees of numerator and denominator), and any holes (common factors that cancel).
📋 Sketch f(x) = (x²−1)/(x−2)
Factor(x−1)(x+1)/(x−2). No common factors → no holes
Vertical asymptotex = 2 (denominator = 0)
End behaviourDegree 2 over degree 1: oblique asymptote. Divide: x+2 + 3/(x−2). Asymptote: y = x+2
x-interceptsx = 1 and x = −1 (numerator = 0)
Quick Sketch Checklist
Complete sketch — 7 steps
1. Domain and any excluded points. 2. Intercepts (x and y). 3. Asymptotes (vertical, horizontal, oblique). 4. f'(x): critical points, sign chart, local extrema. 5. f''(x): inflection points, concavity. 6. End behaviour. 7. Sketch using all collected information.
AM
Dr. Aisha Malik, PhD Mathematics
Senior Lecturer in Applied Mathematics · 12 years teaching calculus
Dr. Malik holds a PhD in Applied Mathematics from the University of Edinburgh and has taught calculus to over 4,000 students. She has reviewed this article for mathematical accuracy and pedagogical clarity.
Technically reviewed by: Prof. James Chen, Stanford Mathematics Department · April 2026