Parabola - Conic Sections and Properties
1️⃣ Understand the definition of a parabola and its components
2️⃣ Master parabola equations in vertex and standard forms
3️⃣ Identify key elements: vertex, focus, directrix, and axis of symmetry
4️⃣ Understand opening direction and determine equation from given information
5️⃣ Apply parabola properties to real-world applications
1️⃣ Definition of a Parabola
A parabola is the locus of all points that are equidistant from a fixed point (focus) and a fixed line (directrix). It is one of the fundamental conic sections with a unique U-shaped curve.
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Distance from any point P to focus = Distance from P to directrix
Key Property: For parabolas, the eccentricity
(exactly), which distinguishes it from ellipses
and hyperbolas
Vertex
The point on the parabola closest to the focus and directrix
Focus
The fixed point used in the definition of the parabola
Directrix
The fixed line used in the definition of the parabola
2️⃣ Parabola Equations and Forms
a) Vertical Parabolas (Opens up or down)
Axis of symmetry is vertical - parabola opens upward or downward
Vertex Form
Standard Form
When
Parabola opens upward (concave up)
When
Parabola opens downward (concave down)
Properties of Vertical Parabolas:
- Vertex:
- Focus:
- Directrix:
- Axis of symmetry:
b) Horizontal Parabolas (Opens left or right)
Axis of symmetry is horizontal - parabola opens leftward or rightward
Horizontal Form
When
Parabola opens rightward
When
Parabola opens leftward
Properties of Horizontal Parabolas:
- Vertex:
- Focus:
- Directrix:
- Axis of symmetry:
3️⃣ Key Relationships
Focus and Directrix Distance
Parameter
Controls opening direction and width
- larger = narrower parabola
- smaller = wider parabola
Vertex
The turning point of the parabola
- Minimum point (if opens up)
- Maximum point (if opens down)
4️⃣ Converting Between Forms
Standard to Vertex Form (Completing the Square)
Example: Convert
Steps:
1. Factor out the coefficient:
2. Complete the square:
3. Add and subtract:
4. Simplify:
5.
Final form:
6. Vertex:
, opens upward
Finding Focus and Directrix
From :
Analysis:
• Vertex:
•
(coefficient), so
• Focus:
• Directrix:
• Axis of symmetry:
5️⃣ Worked Examples
Example 1: Find equation from vertex and focus
Given: Vertex
and Focus
Solution:
• Vertex
• Focus at
means vertical parabola
• Distance from vertex to focus:
• So
• Opens upward (focus above vertex)
•
Equation:
Example 2: Horizontal parabola
Given:
Analysis:
• Vertex:
•
(negative, so opens left)
•
• Focus:
• Directrix:
• Axis of symmetry:
6️⃣ Real-World Applications
Engineering & Architecture
- Suspension bridge cables
- Parabolic reflectors (headlights, satellite dishes)
- Arch structures and domes
- Water fountain trajectories
Physics & Nature
- Projectile motion paths
- Radio telescope design
- Solar concentrators
- Path of thrown objects
Example: Projectile Motion
When a ball is thrown, its path follows a parabolic trajectory due to gravity. The equation
describes the height
as a function of horizontal distance
, where
is gravity,
is initial velocity,
is launch angle, and
is initial height.
7️⃣ Problem-Solving Tips
💡 Strategy for Parabola Problems
- Identify orientation: Vertical or horizontal
- Find the vertex: From graph or by completing the square
- Determine opening: Sign of coefficient
- Calculate focus and directrix: Use distance formula
- Write equation: Choose appropriate form (vertex or standard)
- Verify: Check that key points satisfy the equation
Key Takeaways
1️⃣ Parabolas are defined by equal distances to focus and directrix
2️⃣ Vertex form:
(vertical) or
(horizontal)
3️⃣ Sign of
determines opening direction
4️⃣ Focus distance:
from vertex
5️⃣ Eccentricity
for all parabolas