Quadratic Curve — Definition, Formula & Examples

A quadratic curve is the U-shaped curve you get when you graph a quadratic function. It is another name for a parabola, opening either upward or downward depending on the leading coefficient.

A quadratic curve is the locus of points $(x, y)$ in the Cartesian plane satisfying $y = ax^2 + bx + c$ where $a \neq 0$ . It is a conic section formed by the intersection of a cone with a plane parallel to the cone's slant side, producing a parabola.

Key Formula

y = ax^2 + bx + c

Where:

$a$ = Leading coefficient; controls direction (up/down) and width of the curve. Must not equal zero.
$b$ = Linear coefficient; shifts the axis of symmetry left or right.
$c$ = Constant term; the y-intercept of the curve.

How It Works

Every quadratic function

y = ax^2 + bx + c

traces out exactly one quadratic curve. When

a > 0

, the curve opens upward and has a minimum point (vertex). When

a < 0

, it opens downward and has a maximum. The vertex lies on the axis of symmetry at

x = -\frac{b}{2a}

. The curve crosses the

x

-axis at zero, one, or two points depending on the discriminant

b^2 - 4ac

Worked Example

Problem: Sketch the key features of the quadratic curve

y = 2x^2 - 8x + 6

Find the axis of symmetry: Use the formula for the axis of symmetry.

x = -\frac{b}{2a} = -\frac{-8}{2(2)} = 2

Find the vertex: Substitute

x = 2

back into the equation to get the

y

-coordinate.

y = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2

Determine direction and y-intercept: Since

a = 2 > 0

, the curve opens upward. The y-intercept is

c = 6

, so the curve passes through

(0, 6)

Answer: The quadratic curve has vertex

(2, -2)

, axis of symmetry

x = 2

, opens upward, and crosses the

y

-axis at

(0, 6)

Why It Matters

Quadratic curves model real-world trajectories like the path of a thrown ball or the arc of a water fountain. In physics, engineering, and economics, recognizing a parabolic shape lets you quickly identify maximum/minimum values and predict behavior.

Common Mistakes

Mistake: Assuming every curved graph is a quadratic curve.

Correction: A quadratic curve is specifically a parabola from a degree-2 polynomial. Curves from higher-degree polynomials, exponentials, or other functions have different shapes and properties.