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.

Distance from any point P to focus = Distance from P to directrix

Key Property: For parabolas, the eccentricity $$e = 1$$ (exactly), which distinguishes it from ellipses $$(e < 1)$$ and hyperbolas $$(e > 1)$$

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

$$y - k = a(x - h)^2$$

where $$(h, k)$$ is the vertex

Standard Form

$$y = ax^2 + bx + c$$

Standard quadratic form

When $$a > 0$$

Parabola opens upward (concave up)

When $$a < 0$$

Parabola opens downward (concave down)

Properties of Vertical Parabolas:

Vertex: $$(h, k)$$
Focus: $$(h, k + \frac{1}{4a})$$
Directrix: $$y = k - \frac{1}{4a}$$
Axis of symmetry: $$x = h$$

b) Horizontal Parabolas (Opens left or right)

Axis of symmetry is horizontal - parabola opens leftward or rightward

Horizontal Form

$$x - h = a(y - k)^2$$

where $$(h, k)$$ is the vertex

When $$a > 0$$

Parabola opens rightward

When $$a < 0$$

Parabola opens leftward

Properties of Horizontal Parabolas:

Vertex: $$(h, k)$$
Focus: $$(h + \frac{1}{4a}, k)$$
Directrix: $$x = h - \frac{1}{4a}$$
Axis of symmetry: $$y = k$$

3️⃣ Key Relationships

Focus and Directrix Distance

$$\text{Distance from vertex to focus} = \text{Distance from vertex to directrix} = \frac{1}{4|a|}$$

Parameter $$a$$

Controls opening direction and width

$$|a|$$ larger = narrower parabola
$$|a|$$ smaller = wider parabola

Vertex $$(h, k)$$

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 $$y = 2x^2 - 8x + 11$$

Steps:

1. Factor out the coefficient: $$y = 2(x^2 - 4x) + 11$$
2. Complete the square: $$x^2 - 4x + 4 = (x - 2)^2$$
3. Add and subtract: $$y = 2(x^2 - 4x + 4 - 4) + 11$$
4. Simplify: $$y = 2(x - 2)^2 - 8 + 11$$
5. Final form: $$y = 2(x - 2)^2 + 3$$
6. Vertex: $$(2, 3)$$, opens upward

Finding Focus and Directrix

From $$y = 2(x - 2)^2 + 3$$:

Analysis:

• Vertex: $$(h, k) = (2, 3)$$
• $$a = 2$$ (coefficient), so $$\frac{1}{4a} = \frac{1}{8}$$
• Focus: $$(2, 3 + \frac{1}{8}) = (2, \frac{25}{8})$$
• Directrix: $$y = 3 - \frac{1}{8} = \frac{23}{8}$$
• Axis of symmetry: $$x = 2$$

5️⃣ Worked Examples

Example 1: Find equation from vertex and focus

Given: Vertex $$(1, -2)$$ and Focus $$(1, 0)$$

Solution:

• Vertex $$(h, k) = (1, -2)$$
• Focus at $$(1, 0)$$ means vertical parabola
• Distance from vertex to focus: $$0 - (-2) = 2$$
• So $$\frac{1}{4a} = 2 \Rightarrow a = \frac{1}{8}$$
• Opens upward (focus above vertex)
• Equation: $$y + 2 = \frac{1}{8}(x - 1)^2$$

Example 2: Horizontal parabola

Given: $$x = -\frac{1}{4}(y - 3)^2 + 2$$

Analysis:

• Vertex: $$(2, 3)$$
• $$a = -\frac{1}{4}$$ (negative, so opens left)
• $$\frac{1}{4|a|} = \frac{1}{4 \cdot \frac{1}{4}} = 1$$
• Focus: $$(2 - 1, 3) = (1, 3)$$
• Directrix: $$x = 2 + 1 = 3$$
• Axis of symmetry: $$y = 3$$

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 $$y = -\frac{g}{2v_0^2\cos^2\theta}x^2 + x\tan\theta + h_0$$ describes the height $$y$$ as a function of horizontal distance $$x$$, where $$g$$ is gravity, $$v_0$$ is initial velocity, $$\theta$$ is launch angle, and $$h_0$$ is initial height.

7️⃣ Problem-Solving Tips

💡 Strategy for Parabola Problems

Identify orientation: Vertical $$(y = ...)$$ or horizontal $$(x = ...)$$
Find the vertex: From graph or by completing the square
Determine opening: Sign of coefficient $$a$$
Calculate focus and directrix: Use $$\frac{1}{4a}$$ 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: $$y - k = a(x - h)^2$$ (vertical) or $$x - h = a(y - k)^2$$ (horizontal)

3️⃣ Sign of $$a$$ determines opening direction

4️⃣ Focus distance: $$\frac{1}{4|a|}$$ from vertex

5️⃣ Eccentricity $$e = 1$$ for all parabolas

Parabola

all about Parabola with Simulation