Curve Sketching Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Curve Sketching.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Using the first and second derivatives to determine a function's behavior: intervals of increase/decrease, local maxima/minima, concavity (up/down), and inflection points, then combining this information to sketch an accurate graph.

The first derivative tells you whether the function goes up or down (like reading a speedometer). The second derivative tells you whether it's speeding up or slowing down (like reading an accelerometer). Together, they give you a complete picture of the curve's shape.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Systematic curve sketching: (1) find domain and intercepts, (2) find f' → critical points → increase/decrease, (3) find f'' → concavity → inflection points, (4) check end behavior and asymptotes, (5) plot key points and sketch.

Common stuck point: An inflection point requires f'' to change sign, not just equal zero. f(x) = x^4 has f''(0) = 0 but no inflection point because f'' doesn't change sign at x = 0.

Sense of Study hint: Make a sign chart for f' and f'' using the critical points, then mark +/- in each interval to see where the function rises, falls, and bends.

Worked Examples

Example 1

easy
Sketch f(x) = x^3 - 3x^2: find critical points, monotonicity, concavity, and inflection points.

Solution

  1. 1
    f'(x) = 3x^2-6x = 3x(x-2). Critical points: x=0, 2.
  2. 2
    Increases on (-\infty,0), decreases on (0,2), increases on (2,\infty).
  3. 3
    Local max (0,0), local min (2,-4).
  4. 4
    f''(x)=6x-6. Zero at x=1; sign changes: inflection at (1,-2).
  5. 5
    Concave down on (-\infty,1), concave up on (1,\infty).

Answer

Local max (0,0); local min (2,-4); inflection (1,-2).
Systematic: find f' for critical points/monotonicity, f'' for concavity/inflection. Use sign charts.

Example 2

hard
Sketch f(x) = \dfrac{x^2}{x^2-1}: find domain, asymptotes, critical points, and concavity.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
For f(x) = x^4 - 4x^3, find and classify all critical points.

Example 2

medium
Find the inflection points of f(x) = x^4 - 6x^2.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

derivativedifferentiation rulesoptimization