# Nakafa Framework: LLM

URL: /en/subject/high-school/12/mathematics/analytic-geometry/parabola
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/12/mathematics/analytic-geometry/parabola/en.mdx

Output docs content for large language models.

---

export const metadata = {
  title: "Parabola",
  description: "Master parabola equations, vertex, focus, and directrix. Learn standard forms, real-world applications, and solve problems step-by-step with examples.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Analytic Geometry",
};

import { getColor } from "@repo/design-system/lib/color";
import { LineEquation } from "@repo/design-system/components/contents/line-equation";

## 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.

<LineEquation
  title={<>Parabola <InlineMath math="y = ax^2" /> in Real Life</>}
  description={<>Visualization of parabola with focus at <InlineMath math="F(0, \frac{1}{4a})" /> and directrix <InlineMath math="y = -\frac{1}{4a}" />.</>}
  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;
        const x = 0;
        return {
          x: x,
          y: yMin + i * (yMax - yMin),
          z: 0,
        };
      }),
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  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** <InlineMath math="O(0,0)" />. There are four basic standard parabola forms depending on the direction of the curve's opening.

<LineEquation
  title={<>Four Standard Parabola Forms with Vertex <InlineMath math="O(0,0)" /></>}
  description={<>Standard parabolas <InlineMath math="x^2 = 4py" /> (vertical) and <InlineMath math="y^2 = 4px" /> (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] }],
    },
    {
      points: [{ x: 0, y: 0, z: 0 }], // vertex of all parabolas at O(0,0)
      color: getColor("ROSE"),
      showPoints: true,
      labels: [{ text: "O(0,0)", at: 0, offset: [-0.8, -0.5, 0] }],
    },
    {
      points: Array.from({ length: 2 }, (_, i) => {
        const xMin = -4;
        const xMax = 4;
        const y = 0;
        return {
          x: xMin + i * (xMax - xMin),
          y: y,
          z: 0,
        };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      smooth: false,
    },
    {
      points: Array.from({ length: 2 }, (_, i) => {
        const yMin = -3;
        const yMax = 3;
        const x = 0;
        return {
          x: x,
          y: yMin + i * (yMax - yMin),
          z: 0,
        };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

The four standard parabola equation forms are:

<div className="flex flex-col gap-4">
  <BlockMath math="y^2 = 4px \quad \text{(opens rightward)}" />
  <BlockMath math="y^2 = -4px \quad \text{(opens leftward)}" />
  <BlockMath math="x^2 = 4py \quad \text{(opens upward)}" />
  <BlockMath math="x^2 = -4py \quad \text{(opens downward)}" />
</div>

The value <InlineMath math="p" /> in these equations represents the **distance from vertex to focus**. The larger the value of <InlineMath math="p" />, the more "open" the parabola becomes.

For example, for the parabola <InlineMath math="y^2 = 4px" /> (opens rightward):

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Vertex: } (0, 0)" />
  <BlockMath math="\text{Focus: } (p, 0)" />
  <BlockMath math="\text{Directrix: } x = -p" />
  <BlockMath math="\text{Axis of symmetry: } X\text{-axis}" />
</div>

> Remember that the sign of <InlineMath math="p" /> 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 <InlineMath math="(h, k)" />. This is the **general form of parabola**.

<LineEquation
  title={<>Parabola <InlineMath math="(x-h)^2 = 4p(y-k)" /> with Vertex <InlineMath math="(h,k)" /></>}
  description={<>Parabola with vertex at <InlineMath math="(2, 1)" /> and parameter <InlineMath math="p = 0.5" />.</>}
  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: "both" },
      labels: [{ text: "Axis of symmetry x = 2", at: 1, offset: [1, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 2 }, (_, i) => {
        const xMin = -3;
        const xMax = 7;
        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;
        const x = 0;
        return {
          x: x,
          y: yMin + i * (yMax - yMin),
          z: 0,
        };
      }),
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

General form of parabola equations with vertex at <InlineMath math="(h, k)" />:

<div className="flex flex-col gap-4">
  <BlockMath math="(y-k)^2 = 4p(x-h) \quad \text{(horizontal)}" />
  <BlockMath math="(x-h)^2 = 4p(y-k) \quad \text{(vertical)}" />
</div>

For vertical parabola <InlineMath math="(x-h)^2 = 4p(y-k)" />:

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Vertex: } (h, k)" />
  <BlockMath math="\text{Focus: } (h, k+p)" />
  <BlockMath math="\text{Directrix: } y = k-p" />
  <BlockMath math="\text{Axis of symmetry: } x = h" />
</div>

For horizontal parabola <InlineMath math="(y-k)^2 = 4p(x-h)" />:

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Vertex: } (h, k)" />
  <BlockMath math="\text{Focus: } (h+p, k)" />
  <BlockMath math="\text{Directrix: } x = h-p" />
  <BlockMath math="\text{Axis of symmetry: } y = k" />
</div>

> The sign of <InlineMath math="p" /> 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 <InlineMath math="x" /> is squared → **vertical** parabola (opens up/down)
- If variable <InlineMath math="y" /> is squared → **horizontal** parabola (opens left/right)

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

**Example**: Given a parabola with equation <InlineMath math="x^2 - 4x - 8y + 12 = 0" />

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

<div className="flex flex-col gap-4">
  <BlockMath math="x^2 - 4x - 8y + 12 = 0" />
  <BlockMath math="x^2 - 4x = 8y - 12" />
  <BlockMath math="x^2 - 4x + 4 = 8y - 12 + 4" />
  <BlockMath math="(x-2)^2 = 8y - 8" />
  <BlockMath math="(x-2)^2 = 8(y-1)" />
</div>

From the form <InlineMath math="(x-2)^2 = 8(y-1)" />, we identify:

- <InlineMath math="h = 2" />, <InlineMath math="k = 1" />
- <InlineMath math="4p = 8" />, so <InlineMath math="p = 2" />

Therefore, the parabola elements are:

<LineEquation
  title={<>Parabola Analysis <InlineMath math="(x-2)^2 = 8(y-1)" /></>}
  description={<>Identifying vertex <InlineMath math="(2,1)" />, focus <InlineMath math="(2,3)" />, directrix <InlineMath math="y = -1" />, and <InlineMath math="p = 2" />.</>}
  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: "both" },
      labels: [{ text: "Axis of symmetry", at: 1, offset: [1, 0.5, 0] }],
    },
    {
      points: Array.from({ length: 2 }, (_, i) => {
        const xMin = -3;
        const xMax = 7;
        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 = -3;
        const yMax = 7;
        const x = 0;
        return {
          x: x,
          y: yMin + i * (yMax - yMin),
          z: 0,
        };
      }),
      color: getColor("ROSE"),
      showPoints: false,
      smooth: false,
    },
  ]}
  cameraPosition={[0, 0, 15]}
  showZAxis={false}
/>

From this analysis we obtain:

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Vertex: } (2, 1)" />
  <BlockMath math="\text{Focus: } (2, 3)" />
  <BlockMath math="\text{Directrix: } y = -1" />
  <BlockMath math="\text{Axis of symmetry: } x = 2" />
  <BlockMath math="\text{Opening direction: upward (since } p = 2 > 0\text{)}" />
</div>

## 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 <InlineMath math="(1, -2)" /> and focus at <InlineMath math="(4, -2)" />.

**Solution**:

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

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Vertex: } (h, k) = (1, -2)" />
  <BlockMath math="\text{Focus: } (h+p, k) = (4, -2)" />
</div>

From the focus condition, we get <InlineMath math="h + p = 4" />. Substituting <InlineMath math="h = 1" />:

<BlockMath math="1 + p = 4 \Rightarrow p = 3" />

The equation for a horizontal parabola is <InlineMath math="(y-k)^2 = 4p(x-h)" />:

<div className="flex flex-col gap-4">
  <BlockMath math="(y-(-2))^2 = 4(3)(x-1)" />
  <BlockMath math="(y+2)^2 = 12(x-1)" />
</div>

**Problem 2**: For parabola <InlineMath math="y^2 - 6y - 4x + 13 = 0" />. Determine the coordinates of the vertex and focus.

**Solution**:

Complete the square for variable <InlineMath math="y" />:

<div className="flex flex-col gap-4">
  <BlockMath math="y^2 - 6y - 4x + 13 = 0" />
  <BlockMath math="y^2 - 6y = 4x - 13" />
  <BlockMath math="y^2 - 6y + 9 = 4x - 13 + 9" />
  <BlockMath math="(y-3)^2 = 4x - 4" />
  <BlockMath math="(y-3)^2 = 4(x-1)" />
</div>

From the form <InlineMath math="(y-3)^2 = 4(x-1)" />, we identify:

<div className="flex flex-col gap-4">
  <BlockMath math="h = 1, \quad k = 3" />
  <BlockMath math="4p = 4 \Rightarrow p = 1" />
</div>

Therefore:

<div className="flex flex-col gap-4">
  <BlockMath math="\text{Vertex: } (h, k) = (1, 3)" />
  <BlockMath math="\text{Focus: } (h+p, k) = (1+1, 3) = (2, 3)" />
</div>