Curve sketching
Curve Sketching
Curve Sketching is a topic in Applications of Differentiation in the California Calculus Standards. It is aligned to Standard 9.0, which requires students to use differentiation to sketch, by hand, the graphs of functions.
Curve sketching combines the first derivative (increase, decrease, and extrema) with the second derivative (concavity and inflection points), along with intercepts and asymptotes, to draw a function by hand.
Theory
Curve sketching combines everything the derivatives tell you: the first derivative gives increasing/decreasing behavior and extrema, the second derivative gives concavity and inflection points, and intercepts plus asymptotes fix the rest of the shape.
To sketch a function by hand, assemble its features from the derivatives and a few algebra facts:
- Intercepts: where the graph meets the axes.
- Increasing/decreasing: the sign of \(f'\), with critical points at \(f'=0\).
- Local extrema: where \(f'\) changes sign.
- Concavity and inflection: the sign of \(f''\), with inflection where \(f''\) changes sign.
- End behavior: horizontal or vertical asymptotes for rational functions.
The two derivative signals used in a sketch:
How to sketch a curve
- Intercepts: set \(x=0\) and \(f(x)=0\).
- First derivative: find critical points and increasing/decreasing intervals.
- Second derivative: find concavity and inflection points.
- Asymptotes: check end behavior, then plot the features and connect smoothly.
Combine \(f'\) and \(f''\).
| \(f'(x)\) | \(=\) | \(3x^2-3\Rightarrow x=\pm 1\) |
| \(f''(x)\) | \(=\) | \(6x\Rightarrow \text{inflection at }0\) |
Local max at \(x=-1\), local min at \(x=1\), inflection at \((0,0)\).
Set \(y=0\) and \(x=0\).
| \(x^2-4=0\) | \(\Rightarrow\) | \(x=\pm 2\) |
| \(f(0)\) | \(=\) | \(-4\) |
\(x\)-intercepts \((\pm 2,0)\); \(y\)-intercept \((0,-4)\).
Denominator zero gives a vertical asymptote; degree of bottom is bigger for the horizontal one.
| \(x-3=0\) | \(\Rightarrow\) | \(x=3\ \text{(vertical)}\) |
| \(\lim_{x\to\infty}f\) | \(=\) | \(0\ \text{(horizontal)}\) |
Always increasing (\(f'>0\)) and always concave down (\(f''<0\)).
The graph rises but flattens — like a square-root or log curve.
Common pitfalls
Frequently asked questions
How do you sketch a curve using calculus?
Find intercepts, use \(f'\) for increasing/decreasing and extrema, use \(f''\) for concavity and inflection, check asymptotes, then connect the features smoothly.
What does the first derivative tell you about a graph?
Where it rises (\(f'>0\)) and falls (\(f'<0\)), and where its turning points are (\(f'=0\) with a sign change).
What does the second derivative tell you about a graph?
Its concavity: up where \(f''>0\), down where \(f''<0\), with inflection points where \(f''\) changes sign.
How do you find the asymptotes of a rational function?
Vertical asymptotes are where the denominator is zero (and the numerator is not); the horizontal asymptote comes from comparing degrees as \(x\to\infty\).