Parabola Calculator
Calculate parabola properties including vertex, focus, directrix, and equation forms
Parabola Properties Calculator
Standard Form: y = ax² + bx + c
Cannot be zero
Parabola Properties
Vertex
(0.0000, 0.0000)
Focus
(0.0000, 0.2500)
Directrix
y = -0.2500
Axis of Symmetry
x = 0.0000
Latus Rectum Length
1.0000
Opens
upward
Equation Forms
y = 1x² +0x +0
y = 1(x -0)² +0
Step-by-Step Calculation
1. Given Equation
y = 1x² +0x +0
2. Find Vertex
h = -b/(2a) = -(0)/(2×1) = 0.0000
k = c - b²/(4a) = 0 - (0)²/(4×1) = 0.0000
Vertex: (0.0000, 0.0000)
3. Calculate Focus
Distance from vertex to focus: p = 1/(4a) = 1/(4×1) = 0.2500
Focus: (h, k + p) = (0.0000, 0.0000 + 0.2500) = (0.0000, 0.2500)
4. Find Directrix
Directrix: y = k - p = 0.0000 - 0.2500 = y = -0.2500
Example Calculation
Standard Form Example
Given: y = 2x² + 3x - 4
Coefficients: a = 2, b = 3, c = -4
Vertex:
h = -3/(2×2) = -0.75
k = -4 - 9/8 = -5.125
Focus: (-0.75, -5.0)
Directrix: y = -5.25
Key Parabola Properties
Vertex
The point where the parabola changes direction
Focus
Point equidistant from all points on the parabola
Directrix
Line equidistant from all points on the parabola
Axis of Symmetry
Line through vertex perpendicular to directrix
Calculator Tips
Coefficient 'a' determines parabola width and direction
Positive 'a' opens up/right, negative opens down/left
Larger |a| values create narrower parabolas
Use vertex form for easier graphing
Understanding Parabolas
What is a Parabola?
A parabola is a U-shaped symmetrical curve that represents the graph of a quadratic function. Its defining property is that every point on the parabola is equidistant from both a fixed point (focus) and a fixed line (directrix).
Key Components
- •Vertex: The turning point of the parabola
- •Focus: Point inside the parabola that defines its shape
- •Directrix: Line outside the parabola that defines its shape
- •Axis of Symmetry: Line through vertex and focus
Equation Forms
Standard Form (Vertical)
y = ax² + bx + c
Used when parabola opens up or down
Vertex Form (Vertical)
y = a(x - h)² + k
Where (h, k) is the vertex
Horizontal Forms
x = ay² + by + c
x = a(y - k)² + h
Used when parabola opens left or right
Real-World Applications
Physics
Projectile motion, satellite dishes, car headlights
Architecture
Arches, bridges, suspension cables
Economics
Profit maximization, cost minimization