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URL: https://nakafa.com/en/subjects/mathematics/analytic-geometry/parabola
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/analytic-geometry/parabola/en.mdx

Learn parabola equations, vertex, focus, and directrix. Learn standard forms, real-world applications, and solve problems one step at a time with examples.

---

## Understanding Parabola

Have you ever thrown a basketball to the hoop? Or seen a water fountain spraying water into the air? The trajectory formed by these movements creates a very special curve in mathematics, called a **parabola**.

A parabola is not just an ordinary curve. Its unique shape makes it very useful in real life. Satellite antennas are parabolic in shape to capture signals, spotlights use parabolic reflectors to focus light, and even bridges and modern architecture often use parabolic arches for their strength.

What's interesting about parabolas is their perfect **reflection** property. All rays that come parallel to the parabola's axis will be reflected toward one point called the **focus**. This is why parabolas are so effective for collecting or transmitting energy.

Component: LineEquation
Props:
- title: Parabola $$y = ax^2$$ in Real Life
  Visible text: Parabola in Real Life
- description: Visualization of parabola with focus at $$F(0, \frac{1}{4a})$$ and directrix $$y = -\frac{1}{4a}$$.
  Visible text: Visualization of parabola with focus at and directrix .
- data: [
{
points: Array.from({ length: 101 }, (_, i) => {
const x = (i - 50) / 10; // x from -5 to 5
const y = 0.2 * x * x; // parabola y = 0.2x²
return { x, y, z: 0 };
}),
color: getColor("PURPLE"),
showPoints: false,
labels: [{ text: "Parabolic Path", at: 80, offset: [1, 1, 0] }],
},
{
points: [{ x: 0, y: 1.25, z: 0 }], // focus at (0, 1/(4*0.2)) = (0, 1.25)
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "Focus", at: 0, offset: [0.5, 0.5, 0] }],
},
{
points: [{ x: 0, y: 0, z: 0 }], // vertex of parabola
color: getColor("CYAN"),
showPoints: true,
labels: [{ text: "Vertex", at: 0, offset: [-0.8, -0.5, 0] }],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const x = -6 + i * 12;
const y = -1.25; // directrix at y = -1/(4*0.2) = -1.25
return { x, y, z: 0 };
}),
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "Directrix", at: 1, offset: [0, -0.5, 0] }],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const xMin = -6;
const xMax = 6;
const y = 0;
return {
x: xMin + i * (xMax - xMin),
y: y,
z: 0,
};
}),
color: getColor("ROSE"),
showPoints: false,
smooth: false,
},
{
points: Array.from({ length: 2 }, (_, i) => {
const yMin = -2;
const yMax = 6; ... [truncated; 1334 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

## Mathematical Definition

Mathematically, **a parabola is the locus of points that are equidistant from a fixed point and a fixed line**. The fixed point is called the **focus**, while the fixed line is called the **directrix**.

Imagine you have one point (focus) and one straight line (directrix). Now find all points whose distance to the focus equals their distance to the directrix. The collection of these points is what forms a parabola!

This definition gives us a systematic way to understand parabolas. No matter what the orientation or position, as long as it satisfies the condition of equal distance to the focus and directrix, then it is a parabola.

> This unique property of parabolas makes every point on the curve have a perfect "balance" of distance between the focus and directrix.

## Standard Parabola Equations

Let's start with the simplest parabola: **a parabola with vertex at the origin** $$O(0,0)$$. There are four basic standard parabola forms depending on the direction of the curve's opening.

Visible text: Let's start with the simplest parabola: **a parabola with vertex at the origin** . There are four basic standard parabola forms depending on the direction of the curve's opening.

Component: LineEquation
Props:
- title: Four Standard Parabola Forms with Vertex $$O(0,0)$$
  Visible text: Four Standard Parabola Forms with Vertex
- description: Standard parabolas $$x^2 = 4py$$ (vertical) and $$y^2 = 4px$$ (horizontal).
  Visible text: Standard parabolas (vertical) and (horizontal).
- data: [
{
points: Array.from({ length: 61 }, (_, i) => {
const x = (i - 30) / 10; // x from -3 to 3
const y = 0.25 * x * x; // parabola x² = 4py with p=0.25
return { x, y, z: 0 };
}),
color: getColor("PURPLE"),
showPoints: false,
labels: [{ text: "x² = 4py (upward)", at: 11, offset: [-3, 2, 0] }],
},
{
points: Array.from({ length: 61 }, (_, i) => {
const x = (i - 30) / 10; // x from -3 to 3
const y = -0.25 * x * x; // parabola x² = -4py with p=0.25
return { x, y, z: 0 };
}),
color: getColor("ORANGE"),
showPoints: false,
labels: [{ text: "x² = -4py (downward)", at: 50, offset: [3, -2.5, 0] }],
},
{
points: Array.from({ length: 61 }, (_, i) => {
const y = (i - 30) / 10; // y from -3 to 3
const x = 0.25 * y * y; // parabola y² = 4px with p=0.25
return { x, y, z: 0 };
}),
color: getColor("CYAN"),
showPoints: false,
labels: [{ text: "y² = 4px (rightward)", at: 50, offset: [3, 1.5, 0] }],
},
{
points: Array.from({ length: 61 }, (_, i) => {
const y = (i - 30) / 10; // y from -3 to 3
const x = -0.25 * y * y; // parabola y² = -4px with p=0.25
return { x, y, z: 0 };
}),
color: getColor("TEAL"),
showPoints: false,
labels: [{ text: "y² = -4px (leftward)", at: 11, offset: [-3, -1.5, 0] }],
},
{
point ... [truncated; 1801 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

The four standard parabola equation forms are:

Component: MathContainer
Children:

```math
y^2 = 4px \quad \text{(opens rightward)}
```

```math
y^2 = -4px \quad \text{(opens leftward)}
```

```math
x^2 = 4py \quad \text{(opens upward)}
```

```math
x^2 = -4py \quad \text{(opens downward)}
```

The value $$p$$ in these equations represents the **distance from vertex to focus**. The larger the value of $$p$$, the more "open" the parabola becomes.

Visible text: The value in these equations represents the **distance from vertex to focus**. The larger the value of , the more "open" the parabola becomes.

For example, for the parabola $$y^2 = 4px$$ (opens rightward):

Visible text: For example, for the parabola (opens rightward):

Component: MathContainer
Children:

```math
\text{Vertex: } (0, 0)
```

```math
\text{Focus: } (p, 0)
```

```math
\text{Directrix: } x = -p
```

```math
\text{Axis of symmetry: } X\text{-axis}
```

> Remember that the sign of $$p$$ determines the opening direction: positive for right/up, negative for left/down.

Visible text: > Remember that the sign of determines the opening direction: positive for right/up, negative for left/down.

## Parabola with Arbitrary Vertex

In real applications, parabolas don't always center at the origin. A parabola can have its vertex at any point $$(h, k)$$. This is the **general form of parabola**.

Visible text: In real applications, parabolas don't always center at the origin. A parabola can have its vertex at any point . This is the **general form of parabola**.

Component: LineEquation
Props:
- title: Parabola $$(x-h)^2 = 4p(y-k)$$ with Vertex $$(h,k)$$
  Visible text: Parabola with Vertex
- description: Parabola with vertex at $$(2, 1)$$ and parameter $$p = 0.5$$.
  Visible text: Parabola with vertex at and parameter .
- data: [
{
points: Array.from({ length: 81 }, (_, i) => {
const x = (i - 40) / 10; // x from -4 to 4
const h = 2, k = 1, p = 0.5;
const y = k + (1/(4*p)) * Math.pow(x - h, 2); // parabola with vertex (h,k)
return { x, y, z: 0 };
}),
color: getColor("INDIGO"),
showPoints: false,
labels: [{ text: "Parabola", at: 60, offset: [3, 1, 0] }],
},
{
points: [{ x: 2, y: 1, z: 0 }], // vertex
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "Vertex (2,1)", at: 0, offset: [-1, -0.2, 0] }],
},
{
points: [{ x: 2, y: 1.5, z: 0 }], // focus at (h, k+p) = (2, 1+0.5)
color: getColor("LIME"),
showPoints: true,
labels: [{ text: "Focus (2,1.5)", at: 0, offset: [0.5, 1, 0] }],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const x = -2 + i * 8;
const y = 0.5; // directrix at y = k-p = 1-0.5 = 0.5
return { x, y, z: 0 };
}),
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "Directrix y = 0.5", at: 1, offset: [0, 0.5, 0] }],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const yMin = -1;
const yMax = 5;
const x = 2; // axis of symmetry
return { x, y: yMin + i * (yMax - yMin), z: 0 };
}),
color: getColor("PINK"),
showPoints: false,
smooth: false,
cone: { position: " ... [truncated; 1713 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

General form of parabola equations with vertex at $$(h, k)$$:

Visible text: General form of parabola equations with vertex at :

Component: MathContainer
Children:

```math
(y-k)^2 = 4p(x-h) \quad \text{(horizontal)}
```

```math
(x-h)^2 = 4p(y-k) \quad \text{(vertical)}
```

For vertical parabola $$(x-h)^2 = 4p(y-k)$$:

Visible text: For vertical parabola :

Component: MathContainer
Children:

```math
\text{Vertex: } (h, k)
```

```math
\text{Focus: } (h, k+p)
```

```math
\text{Directrix: } y = k-p
```

```math
\text{Axis of symmetry: } x = h
```

For horizontal parabola $$(y-k)^2 = 4p(x-h)$$:

Visible text: For horizontal parabola :

Component: MathContainer
Children:

```math
\text{Vertex: } (h, k)
```

```math
\text{Focus: } (h+p, k)
```

```math
\text{Directrix: } x = h-p
```

```math
\text{Axis of symmetry: } y = k
```

> The sign of $$p$$ determines the parabola's opening direction. Positive for up/right, negative for down/left.

Visible text: > The sign of determines the parabola's opening direction. Positive for up/right, negative for down/left.

## Determining Parabola Elements

When given a parabola equation, we can identify all its important elements systematically. The first step is to **recognize the parabola's orientation**:

- If variable $$x$$ is squared → **vertical** parabola (opens up/down)
- If variable $$y$$ is squared → **horizontal** parabola (opens left/right)

Visible text: - If variable is squared → **vertical** parabola (opens up/down)
- If variable is squared → **horizontal** parabola (opens left/right)

Let's see how to do this with an example.

**Example**: Given a parabola with equation $$x^2 - 4x - 8y + 12 = 0$$

Visible text: **Example**: Given a parabola with equation

The first step is to convert to standard form by completing the square:

Component: MathContainer
Children:

```math
x^2 - 4x - 8y + 12 = 0
```

```math
x^2 - 4x = 8y - 12
```

```math
x^2 - 4x + 4 = 8y - 12 + 4
```

```math
(x-2)^2 = 8y - 8
```

```math
(x-2)^2 = 8(y-1)
```

From the form $$(x-2)^2 = 8(y-1)$$, we identify:

Visible text: From the form , we identify:

- $$h = 2$$, $$k = 1$$
- $$4p = 8$$, so $$p = 2$$

Visible text: - , 
- , so

Therefore, the parabola elements are:

Component: LineEquation
Props:
- title: Parabola Analysis $$(x-2)^2 = 8(y-1)$$
  Visible text: Parabola Analysis
- description: Identifying vertex $$(2,1)$$, focus $$(2,3)$$, directrix $$y = -1$$, and $$p = 2$$.
  Visible text: Identifying vertex , focus , directrix , and .
- data: [
{
points: Array.from({ length: 81 }, (_, i) => {
const x = (i - 40) / 8; // x from -5 to 5
const h = 2, k = 1, p = 2;
const y = k + (1/(4*p)) * Math.pow(x - h, 2); // (x-h)² = 4p(y-k)
return { x, y, z: 0 };
}),
color: getColor("VIOLET"),
showPoints: false,
labels: [{ text: "(x-2)² = 8(y-1)", at: 60, offset: [2, 1.5, 0] }],
},
{
points: [{ x: 2, y: 1, z: 0 }], // vertex (h,k)
color: getColor("ORANGE"),
showPoints: true,
labels: [{ text: "Vertex (2,1)", at: 0, offset: [-1, -0.5, 0] }],
},
{
points: [{ x: 2, y: 3, z: 0 }], // focus (h, k+p) = (2, 1+2)
color: getColor("LIME"),
showPoints: true,
labels: [{ text: "Focus (2,3)", at: 0, offset: [0.8, 0.5, 0] }],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const x = -2 + i * 8;
const y = -1; // directrix y = k-p = 1-2 = -1
return { x, y, z: 0 };
}),
color: getColor("AMBER"),
showPoints: false,
smooth: false,
labels: [{ text: "Directrix y = -1", at: 1, offset: [0, -0.5, 0] }],
},
{
points: Array.from({ length: 2 }, (_, i) => {
const yMin = -2;
const yMax = 6;
const x = 2; // axis of symmetry x = h = 2
return { x, y: yMin + i * (yMax - yMin), z: 0 };
}),
color: getColor("PINK"),
showPoints: false,
smooth: false,
cone: { position: "bo ... [truncated; 1705 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

From this analysis we obtain:

Component: MathContainer
Children:

```math
\text{Vertex: } (2, 1)
```

```math
\text{Focus: } (2, 3)
```

```math
\text{Directrix: } y = -1
```

```math
\text{Axis of symmetry: } x = 2
```

```math
\text{Opening direction: upward (since } p = 2 > 0\text{)}
```

## Example Problems

Let's work through some problems to deepen our understanding of parabolas.

**Problem** $$1$$: Determine the equation of a parabola that has vertex at $$(1, -2)$$ and focus at $$(4, -2)$$.

Visible text: **Problem** : Determine the equation of a parabola that has vertex at and focus at .

**Solution**:

Since the vertex and focus have the same $$y$$-coordinate, this parabola is **horizontal**.

Visible text: Since the vertex and focus have the same -coordinate, this parabola is **horizontal**.

Component: MathContainer
Children:

```math
\text{Vertex: } (h, k) = (1, -2)
```

```math
\text{Focus: } (h+p, k) = (4, -2)
```

From the focus condition, we get $$h + p = 4$$. Substituting $$h = 1$$:

Visible text: From the focus condition, we get . Substituting :

```math
1 + p = 4 \Rightarrow p = 3
```

The equation for a horizontal parabola is $$(y-k)^2 = 4p(x-h)$$:

Visible text: The equation for a horizontal parabola is :

Component: MathContainer
Children:

```math
(y-(-2))^2 = 4(3)(x-1)
```

```math
(y+2)^2 = 12(x-1)
```

**Problem** $$2$$: For parabola $$y^2 - 6y - 4x + 13 = 0$$. Determine the coordinates of the vertex and focus.

Visible text: **Problem** : For parabola . Determine the coordinates of the vertex and focus.

**Solution**:

Complete the square for variable $$y$$:

Visible text: Complete the square for variable :

Component: MathContainer
Children:

```math
y^2 - 6y - 4x + 13 = 0
```

```math
y^2 - 6y = 4x - 13
```

```math
y^2 - 6y + 9 = 4x - 13 + 9
```

```math
(y-3)^2 = 4x - 4
```

```math
(y-3)^2 = 4(x-1)
```

From the form $$(y-3)^2 = 4(x-1)$$, we identify:

Visible text: From the form , we identify:

Component: MathContainer
Children:

```math
h = 1, \quad k = 3
```

```math
4p = 4 \Rightarrow p = 1
```

Therefore:

Component: MathContainer
Children:

```math
\text{Vertex: } (h, k) = (1, 3)
```

```math
\text{Focus: } (h+p, k) = (1+1, 3) = (2, 3)
```