Parabola (geometry)
A parabola is a symmetric open curve formed by the intersection of a cone with a plane parallel to one of its sides, defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix). It is a conic section with eccentricity exactly equal to 1, placing it between the ellipse and the hyperbola. Parabolas appear extensively in physics (projectile paths, parabolic mirrors, antenna reflectors) and engineering design.
Key Formula
y = ax² + bx + c
LaTeX: y = ax^2 + bx + c
| Symbol | Meaning | Unit |
|---|---|---|
| a | coefficient determining width and direction of opening | dimensionless |
| b | coefficient determining horizontal shift | dimensionless |
| c | y-intercept (constant term) | dimensionless |
| x | independent variable (horizontal axis) | dimensionless |
| y | dependent variable (vertical axis) | dimensionless |
Worked Example
Problem
A parabolic arch has the equation y = -x² + 4x, where x and y are in metres. Find the vertex (maximum height) and the width at ground level.
Solution
Step 1: Rewrite in vertex form by completing the square. y = -(x² - 4x) = -[(x - 2)² - 4] = -(x - 2)² + 4 Step 2: Vertex is at (h, k) = (2, 4), so maximum height = 4 m at x = 2 m. Step 3: Find roots (y = 0): -(x - 2)² + 4 = 0 → (x - 2)² = 4 → x - 2 = ±2 → x = 0 or x = 4. Step 4: Width at ground level = 4 - 0 = 4 m.
Answer
Maximum height = 4 m at x = 2 m; arch width at ground level = 4 m.
Standard Forms of a Parabola
| Orientation | Standard Form | Vertex | Focus | Directrix |
|---|---|---|---|---|
| Upward (a > 0) | y = a(x − h)² + k | (h, k) | (h, k + 1/(4a)) | y = k − 1/(4a) |
| Downward (a < 0) | y = a(x − h)² + k | (h, k) | (h, k − 1/(4|a|)) | y = k + 1/(4|a|) |
| Rightward | x = a(y − k)² + h | (h, k) | (h + 1/(4a), k) | x = h − 1/(4a) |
| Leftward | x = a(y − k)² + h | (h, k) | (h − 1/(4|a|), k) | x = h + 1/(4|a|) |
| General conic | Ax² + Bxy + Cy² + Dx + Ey + F = 0 | Variable | B² − 4AC = 0 | Eccentricity e = 1 |
Interactive Tools
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Related Terms
From Greek parabolē (παραβολή), meaning "application" or "comparison", from para- ("beside") + bolē ("throw"). The term was coined by Apollonius of Perga (c. 262–190 BC) in his landmark work Conics, because the square on the ordinate equals the rectangle "applied" to the latus rectum.