Cubic Curve — Definition, Formula & Examples

A cubic curve is a curve defined by a polynomial equation of degree 3. Its graph can have up to two turning points and one inflection point, giving it a characteristic S-shape or N-shape.

A cubic curve is the set of all points $(x, y)$ in the plane satisfying a polynomial equation of total degree 3. In the simplest case of a cubic function $y = ax^3 + bx^2 + cx + d$ where $a \neq 0$ , the curve is the graph of a single-variable cubic polynomial. More generally, a cubic curve can be defined by any equation $P(x, y) = 0$ where $P$ is a polynomial of total degree 3.

Key Formula

y = ax^3 + bx^2 + cx + d, \quad a \neq 0

Where:

$a$ = Leading coefficient controlling end behavior and steepness
$b$ = Coefficient of the quadratic term
$c$ = Coefficient of the linear term
$d$ = Constant term (y-intercept)

How It Works

The shape of a basic cubic

y = ax^3 + bx^2 + cx + d

depends on its coefficients. When

a > 0

, the curve rises from left to right overall; when

a < 0

, it falls. The discriminant

\Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2

determines whether the cubic has one real root or three distinct real roots. Every cubic curve has exactly one inflection point, where the concavity changes from concave up to concave down or vice versa. Finding this inflection point by setting the second derivative equal to zero is a standard calculus exercise.

Worked Example

Problem: Find the inflection point and turning points of the cubic curve y = x³ − 3x² + 2.

Find the first derivative: Differentiate to find where the slope is zero (turning points).

y' = 3x^2 - 6x = 3x(x - 2)

Solve for turning points: Set y' = 0 to get x = 0 and x = 2. Evaluate y at each: y(0) = 2 and y(2) = 8 − 12 + 2 = −2.

\text{Local max: }(0,\, 2) \qquad \text{Local min: }(2,\, -2)

Find the inflection point: Take the second derivative and set it to zero.

y'' = 6x - 6 = 0 \implies x = 1,\quad y(1) = 1 - 3 + 2 = 0

Answer: The cubic has a local maximum at (0, 2), a local minimum at (2, −2), and an inflection point at (1, 0) where the concavity changes.

Why It Matters

Cubic curves appear in physics when modeling motion with non-constant acceleration, in economics for cost functions with changing marginal returns, and in computer graphics for Bézier curves used to draw smooth paths. Understanding their shape is essential preparation for AP Calculus, where analyzing turning points and inflection points of polynomials is a core skill.

Common Mistakes

Mistake: Assuming every cubic has two turning points.

Correction: Some cubics, like y = x³, have no turning points at all — the first derivative may never equal zero or may touch zero without changing sign. The discriminant of the derivative determines whether there are zero or two turning points.