Introduction
A parabola is a U-shaped curve that represents a quadratic function. It’s one of the fundamental conic sections in mathematics, with applications ranging from physics (projectile motion) to engineering (satellite dishes). This comprehensive guide provides all essential parabola formulas organized for easy reference.
Standard Parabola Equations and Properties
| Category | Formula | Description | Variables |
|---|---|---|---|
| Standard Form | y = ax² + bx + c |
General quadratic equation | a ≠ 0, a,b,c are constants |
| Vertex Form | y = a(x - h)² + k |
Parabola with vertex at (h,k) | (h,k) = vertex coordinates |
| Focus-Directrix Form | (x - h)² = 4p(y - k) |
Vertical parabola with focus-directrix relationship | p = focal parameter |
| Focus-Directrix Form | (y - k)² = 4p(x - h) |
Horizontal parabola with focus-directrix relationship | p = focal parameter |
Vertex and Key Points
| Property | Formula | Description | Application |
|---|---|---|---|
| Vertex (x-coordinate) | x = -b/(2a) |
x-coordinate of vertex for y = ax² + bx + c | Finding maximum/minimum point |
| Vertex (y-coordinate) | y = c - b²/(4a) |
y-coordinate of vertex | Completing the vertex calculation |
| Vertex (alternative) | y = f(-b/(2a)) |
Substitute x-vertex into original equation | Alternative calculation method |
| Y-intercept | y = c |
Point where parabola crosses y-axis | Set x = 0 in standard form |
| X-intercepts (Roots) | x = (-b ± √(b² - 4ac))/(2a) |
Quadratic formula for x-intercepts | Solutions to ax² + bx + c = 0 |
Focus, Directrix, and Geometric Properties
| Property | Formula | Description | Notes |
|---|---|---|---|
| Focus (Vertical Parabola) | F(h, k + p) |
Focus point for (x-h)² = 4p(y-k) | p > 0: opens upward; p < 0: opens downward |
| Focus (Horizontal Parabola) | F(h + p, k) |
Focus point for (y-k)² = 4p(x-h) | p > 0: opens rightward; p < 0: opens leftward |
| Directrix (Vertical) | y = k - p |
Directrix line for vertical parabola | Horizontal line below/above vertex |
| Directrix (Horizontal) | x = h - p |
Directrix line for horizontal parabola | Vertical line left/right of vertex |
| Focal Length | p = 1/(4a) |
Distance from vertex to focus | For y = ax² + bx + c form |
| Axis of Symmetry | x = h or x = -b/(2a) |
Line of symmetry through vertex | Vertical line for standard parabolas |
Distance and Length Formulas
| Measurement | Formula | Description | Usage |
|---|---|---|---|
| Focal Distance | ` | p | ` |
| Latus Rectum | `4 | p | ` |
| Arc Length | L = ∫[a to b] √(1 + (dy/dx)²) dx |
Length of parabolic arc from x = a to x = b | Requires calculus integration |
| Arc Length (Simplified) | L = ∫[a to b] √(1 + 4a²x²) dx |
For parabola y = ax² | Specific case calculation |
Area Calculations
| Area Type | Formula | Description | Application |
|---|---|---|---|
| Area Under Curve | A = ∫[a to b] (ax² + bx + c) dx |
Area between parabola and x-axis | Definite integration |
| Area Between Parabola and Line | `A = ∫[a to b] | f(x) – g(x) | dx` |
| Parabolic Segment Area | A = (2/3)bh |
Area of segment cut by a chord | b = chord length, h = height of segment |
Tangent and Normal Lines
| Line Type | Formula | Description | Variables |
|---|---|---|---|
| Slope at Point | dy/dx = 2ax + b |
Derivative gives slope at any point | For y = ax² + bx + c |
| Tangent Line Equation | y - y₁ = m(x - x₁) |
Tangent line at point (x₁, y₁) | m = slope at that point |
| Tangent Line (Vertex Form) | y - k = 2a(x₁ - h)(x - x₁) + a(x₁ - h)² |
Tangent at (x₁, y₁) for vertex form | More complex but useful |
| Normal Line Slope | m_normal = -1/(2ax₁ + b) |
Slope of normal line (perpendicular to tangent) | Negative reciprocal of tangent slope |
Parametric and Polar Forms
| Form | Formula | Description | Parameter Range |
|---|---|---|---|
| Parametric Equations | x = at², y = 2at |
Standard parametric form | t ∈ ℝ (all real numbers) |
| Parametric (General) | x = h + at², y = k + 2at |
Parametric with vertex (h,k) | t ∈ ℝ |
| Polar Form | r = 2p/(1 - cos θ) |
Polar equation with focus at origin | θ ∈ [0, 2π), p > 0 |
Reflection and Optical Properties
| Property | Formula | Description | Physical Application |
|---|---|---|---|
| Reflective Property | All rays parallel to axis reflect through focus | Geometric property of parabolas | Satellite dishes, car headlights |
| Focal Radius | r = y + p |
Distance from any point to focus (vertical parabola) | For point (x, y) on parabola |
Transformation Formulas
| Transformation | Formula | Description | Effect |
|---|---|---|---|
| Vertical Translation | y = a(x - h)² + k |
Moving up/down by k units | Changes vertex position |
| Horizontal Translation | y = a(x - h)² + k |
Moving left/right by h units | Changes vertex position |
| Vertical Stretch/Compression | y = a·f(x) |
Multiply y-values by factor a | |
| Reflection | y = -f(x) |
Reflect across x-axis | Opens downward instead of upward |
Discriminant and Nature of Roots
| Discriminant Value | Formula | Nature of Roots | Graphical Meaning |
|---|---|---|---|
| Δ > 0 | Δ = b² - 4ac > 0 |
Two distinct real roots | Parabola intersects x-axis at two points |
| Δ = 0 | Δ = b² - 4ac = 0 |
One repeated real root | Parabola touches x-axis at vertex |
| Δ < 0 | Δ = b² - 4ac < 0 |
No real roots (complex roots) | Parabola doesn’t intersect x-axis |
Special Case Formulas
| Special Case | Formula | When to Use | Notes |
|---|---|---|---|
| Parabola Through Origin | y = ax² |
When vertex is at (0,0) | Simplest form |
| Symmetric About Y-axis | y = ax² + c |
No linear term (b = 0) | Vertex on y-axis |
| Unit Parabola | y = x² |
Standard reference parabola | a = 1, vertex at origin |
Relationships and Identities
| Relationship | Formula | Description | Significance |
|---|---|---|---|
| Focus-Directrix Definition | Distance to focus = Distance to directrix | Fundamental definition of parabola | Geometric construction principle |
| Vertex-Focus Relationship | p = 1/(4a) |
Connects algebraic and geometric forms | Links standard form to geometric properties |
| Latus Rectum Property | Length = `4 | p | ` |
Summary for Students
Essential Formulas to Remember:
- Standard Form:
y = ax² + bx + c - Vertex:
(-b/(2a), f(-b/(2a))) - Quadratic Formula:
x = (-b ± √(b² - 4ac))/(2a) - Focus:
F(h, k + 1/(4a))for vertex form - Axis of Symmetry:
x = -b/(2a)
Study Tips:
- Practice converting between different forms
- Understand the geometric meaning of each parameter
- Use graphing to visualize the relationships
- Remember that parabolas have reflective properties used in real-world applications
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Parabola FAQs: Definition, Equation, Vertex & Applications
Parabola FAQs
Quick, student-friendly answers to the most searched questions about parabolas—perfect for revision and SEO.
Q. What is a parabola in mathematics?
A parabola is a U-shaped curve formed by the graph of a quadratic equation. It is the set of all points equidistant from a fixed point (the focus) and a fixed straight line (the directrix). Parabolas are common in algebra, coordinate geometry, and real-life designs like satellite dishes and headlights.
Q. What is the standard equation of a parabola?
The standard equation depends on orientation. For an up/down parabola: y = ax² + bx + c. For left/right: x = ay² + by + c. In vertex form: (x − h)² = 4a(y − k) or (y − k)² = 4a(x − h), where (h, k) is the vertex.
Q. What are the main parts of a parabola?
The main parts include the vertex (turning point), axis of symmetry (line dividing the parabola), focus (fixed point), directrix (fixed line), and latus rectum (chord through the focus perpendicular to the axis).
Q. How do you find the vertex of a parabola?
For y = ax² + bx + c, the x-coordinate of the vertex is x = −b/(2a). Substitute this x-value into the equation to get the y-coordinate. The vertex is the maximum point if a < 0 and the minimum point if a > 0.
Q. What determines whether a parabola opens up, down, left, or right?
For y = ax² + bx + c: if a > 0 the parabola opens upward, and if a < 0 it opens downward. For x = ay² + by + c: if a > 0 it opens right, and if a < 0 it opens left.
Q. Where are parabolas used in real life?
Parabolas are used in satellite dishes, antennas, car headlights, flashlights, projectile motion in physics, suspension bridge design, and reflective mirrors in telescopes because of their strong focusing/reflecting properties.
This comprehensive guide covers all fundamental parabola formulas essential for academic success in algebra, precalculus, calculus, and applied mathematics courses.