QUADRATICS

Polynomials of degree 2. Escaping the straight line to model gravity, area, and parabolic curves.

01 // The Parabola

Unlike a line which travels in one direction forever, a parabola is a U-shaped curve. The introduction of the $x^2$ term forces the line to turn around, creating perfect symmetry.

VERTEX

(h, k)

The turning point. The absolute Maximum or Minimum.

AXIS OF SYMMETRYThe invisible mirror line down the center:

x = h

ROOTS (Zeros)Where the parabola crosses the x-axis (

y = 0

VERTEX

02 // The Lab

Warping the Curve

A parabola's shape and position are entirely controlled by three variables. Use the Vertex Form constructor below to dynamically build and shift parabolas across the coordinate plane.

Parabola Constructor

Vertex Form

y = 1(x - 0)² - 5

Vertex: (0, -5)

'a' (Width & Direction)1

'h' (Horizontal Shift)0

'k' (Vertical Shift)-5

15px / Unit

03 // The Three Forms

Default

Standard Form

y = ax^2 + bx + c

Best for determining the y-intercept ( $c$ ) and the direction of opening ( $a$ ). Hard to graph directly.

Graphing

Vertex Form

y = a(x - h)^2 + k

The ultimate visualizer. The peak (or valley) of the curve is visible immediately at coordinate $(h, k)$ .

Solving

Intercept Form

y = a(x - p)(x - q)

The factored state. Best for finding roots. The curve crosses the x-axis exactly at $p$ and $q$ .

The Universal Solvent

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

When standard factoring fails, the Quadratic Formula is guaranteed to solve ANY quadratic equation. It calculates the exact roots by finding the axis of symmetry and adding/subtracting the discriminant spread.

Verification Protocol

Knowledge Check: Parabolas1 / 3

Match the quadratic form to what it reveals most easily.

1. Select Term

2. Assign Match

Curve Dynamics Mastered

You are ready to break down complex polynomials.

Next: Factoring