The shape of a quadratic
The graph of y = ax² + bx + c, with a ≠ 0, is a parabola: a symmetric U-shaped curve that opens upward when a > 0 and downward when a < 0. Parabolas describe the path of a thrown ball, the cables of a suspension bridge under uniform load and the cross-section of satellite dishes.
Key features and formulas
- Vertex: (h, k), with h = −b ÷ (2a) and k = c − b² ÷ (4a)
- Axis of symmetry: the vertical line x = h
- Focal length: p = 1 ÷ (4a)
- Focus: (h, k + p)
- Directrix: the horizontal line y = k − p
- Latus rectum: the chord through the focus, of length |1 ÷ a| = 4|p|
- y-intercept: (0, c)
- x-intercepts: x = (−b ± √(b² − 4ac)) ÷ (2a), when the discriminant b² − 4ac is not negative
The parabola can also be written in vertex form: y = a(x − h)² + k.
Example
For y = x² − 4x + 3:
- Vertex: x = 4 ÷ 2 = 2, y = 3 − 16 ÷ 4 = −1
- Axis of symmetry: x = 2
- Focal length: 1 ÷ 4 = 0.25
- Focus: (2, −0.75); directrix: y = −1.25
- Roots: x = 1 and x = 3
- y-intercept: 3
Focus and directrix
Every point on a parabola is exactly as far from the focus as from the directrix. This definition gives the parabola its reflective property: rays that arrive parallel to the axis all bounce off the curve through the focus. Satellite dishes and radio telescopes concentrate signals at a receiver placed at the focus, and headlights and flashlights do the reverse, sending light from a bulb at the focus out in a parallel beam.
Tips
- A larger |a| makes a narrower parabola, with the focus closer to the vertex.
- If the discriminant is negative, the parabola does not cross the x-axis; the quadratic equation has complex roots.
- To find the equation of a parabola through three points, solve the system for a, b and c with a linear system calculator.