# Nakafa Framework: LLM

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

Output docs content for large language models.

---

export const metadata = {
  title: "Rotation",
  description: "Master function rotation around fixed points with step-by-step examples. Learn rotation formulas, angles, and transformations for various function types.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "05/26/2025",
  subject: "Function Transformation",
};

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

## Understanding Function Rotation

Rotation is a geometric transformation that rotates an object around a specific center point with a determined rotation angle. In the context of functions, rotation changes the position of the function graph by rotating every point on the graph.

Imagine rotating a wheel. Every point on the wheel will move following a circle with the rotation center as its center. Similarly with function graphs, every point will rotate following the same pattern.

## Rotation Formula Around Center Point

For rotation around center point <InlineMath math="(a, b)" /> with angle <InlineMath math="\theta" />, the transformation formula is:

<div className="flex flex-col gap-4">
<BlockMath math="x' = (x - a) \cos \theta - (y - b) \sin \theta + a" />
<BlockMath math="y' = (x - a) \sin \theta + (y - b) \cos \theta + b" />
</div>

Where:
- <InlineMath math="(x, y)" /> is the original point coordinate
- <InlineMath math="(x', y')" /> is the rotated point coordinate
- <InlineMath math="(a, b)" /> is the rotation center point
- <InlineMath math="\theta" /> is the rotation angle (positive for counterclockwise direction)

## Special Rotation Around Origin

When rotation is performed around the origin <InlineMath math="(0, 0)" />, the formula becomes simpler:

1. 90 Degree Rotation

    <BlockMath math="(x, y) \rightarrow (-y, x)" />

2. 180 Degree Rotation

    <BlockMath math="(x, y) \rightarrow (-x, -y)" />

3. 270 Degree Rotation

    <BlockMath math="(x, y) \rightarrow (y, -x)" />

## Visualization of Quadratic Function Rotation

Let's see how rotation affects the quadratic function <InlineMath math="y = x^2" />:

<LineEquation
  title={<>Rotation of Function <InlineMath math="y = x^2" /> Around Origin</>}
  description="Comparison of original function with 90° counterclockwise rotation result."
  data={[
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        const y = x * x * 0.25;
        return { x, y, z: 0 };
      }),
      color: getColor("PURPLE"),
      showPoints: false,
      labels: [
        {
          text: "y = x²",
          at: 15,
          offset: [0.5, 0.5, 0],
        },
      ],
    },
    {
      points: Array.from({ length: 21 }, (_, i) => {
        const x = (i - 10) * 0.5;
        const y = x * x * 0.25;
        // 90° rotation around origin: (x,y) → (-y,x)
        const xRotated = -y;
        const yRotated = x;
        return { x: xRotated, y: yRotated, z: 0 };
      }),
      color: getColor("ORANGE"),
      showPoints: false,
      labels: [
        {
          text: "90° Rotation Result",
          at: 5,
          offset: [-1, 0.5, 0],
        },
      ],
    },
  ]}
/>

## Properties of Function Rotation

Rotation has several important properties:

- **Preserves Shape**: Rotation does not change the basic shape of the graph, only changes its orientation
- **Preserves Distance**: The distance between two points on the graph remains the same after rotation
- **Rotation Composition**: Consecutive rotations can be combined by adding their angles

## Application of Rotation to Various Functions

### Linear Function Rotation

For linear function <InlineMath math="y = mx + c" />, rotation will change the slope and position of the line.

<LineEquation
  title={<>Rotation of Linear Function <InlineMath math="y = 2x + 1" /></>}
  description="45° rotation around origin."
  data={[
    {
      points: Array.from({ length: 11 }, (_, i) => {
        const x = (i - 5);
        const y = 2 * x + 1;
        return { x, y, z: 0 };
      }),
      color: getColor("TEAL"),
      showPoints: false,
      labels: [
        {
          text: "y = 2x + 1",
          at: 6,
          offset: [1, 0.5, 0],
        },
      ],
    },
    {
      points: Array.from({ length: 11 }, (_, i) => {
        const x = (i - 5);
        const y = 2 * x + 1;
        // 45° rotation around origin
        const cos45 = Math.cos(Math.PI / 4);
        const sin45 = Math.sin(Math.PI / 4);
        const xRotated = x * cos45 - y * sin45;
        const yRotated = x * sin45 + y * cos45;
        return { x: xRotated, y: yRotated, z: 0 };
      }),
      color: getColor("VIOLET"),
      showPoints: false,
      labels: [
        {
          text: "45° Rotation",
          at: 6,
          offset: [-0.5, 0.5, 0],
        },
      ],
    },
  ]}
/>

### Exponential Function Rotation

Rotation can also be applied to exponential functions with interesting results.

<LineEquation
  title={<>Rotation of Function <InlineMath math="y = 2^x" /></>}
  description="90° counterclockwise rotation."
  cameraPosition={[8, 6, 8]}
  data={[
    {
      points: Array.from({ length: 15 }, (_, i) => {
        const x = (i - 7) * 0.5;
        const y = Math.pow(2, x * 0.5);
        return { x, y, z: 0 };
      }),
      color: getColor("EMERALD"),
      showPoints: false,
      labels: [
        {
          text: "y = 2^x",
          at: 12,
          offset: [0.5, 0.5, 0],
        },
      ],
    },
    {
      points: Array.from({ length: 15 }, (_, i) => {
        const x = (i - 7) * 0.5;
        const y = Math.pow(2, x * 0.5);
        // 90° rotation around origin: (x,y) → (-y,x)
        const xRotated = -y;
        const yRotated = x;
        return { x: xRotated, y: yRotated, z: 0 };
      }),
      color: getColor("AMBER"),
      showPoints: false,
      labels: [
        {
          text: "90° Rotation",
          at: 3,
          offset: [-0.5, 0.5, 0],
        },
      ],
    },
  ]}
/>

## Steps to Determine Rotation Results

To determine the rotation result of a function, follow these steps:

- Determine the rotation center point and desired rotation angle
- Select several points on the original function graph as samples
- Apply the rotation formula to each sample point
- Connect the rotation result points to form a new function graph
- Verify the result by checking several additional points

## Exercises

1. Determine the rotation result of point <InlineMath math="(3, 4)" /> around the origin with a 90° counterclockwise angle.

2. Function <InlineMath math="f(x) = x^2 - 2x + 1" /> is rotated 180° around the origin. Determine the coordinates of the rotation result vertex if the original vertex is at <InlineMath math="(1, 0)" />.

3. Line <InlineMath math="y = 3x - 2" /> is rotated 270° around the origin. Determine the equation of the rotation result line.

4. Point <InlineMath math="(2, 5)" /> is rotated 60° around point <InlineMath math="(1, 1)" />. Determine the rotation result coordinates.

5. Function <InlineMath math="y = \sqrt{x}" /> for <InlineMath math="x \geq 0" /> is rotated 90° counterclockwise around the origin. Explain the shape of the rotation result graph.

### Answer Key

1. Using the 90° rotation formula:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="(x, y) \rightarrow (-y, x)" />
   <BlockMath math="(3, 4) \rightarrow (-4, 3)" />
   </div>
   
   So the rotation result is <InlineMath math="(-4, 3)" />.

   <LineEquation
     title="Visualization of Point (3, 4) Rotation"
     description="90° counterclockwise rotation around origin."
     cameraPosition={[8, 6, 8]}
     data={[
       {
         points: [
           { x: 0, y: 0, z: 0 },
           { x: 3, y: 4, z: 0 }
         ],
         color: getColor("PURPLE"),
         showPoints: false,
         labels: [
           {
             text: "(3, 4)",
             at: 1,
             offset: [0.3, 0.3, 0],
           },
         ],
         cone: { position: "end", size: 0.3 }
       },
       {
         points: [
           { x: 0, y: 0, z: 0 },
           { x: -4, y: 3, z: 0 }
         ],
         color: getColor("ORANGE"),
         showPoints: false,
         labels: [
           {
             text: "(-4, 3)",
             at: 1,
             offset: [-0.5, 0.3, 0],
           },
         ],
         cone: { position: "end", size: 0.3 }
       },
     ]}
   />

2. Original vertex rotated 180°:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="(x, y) \rightarrow (-x, -y)" />
   <BlockMath math="(1, 0) \rightarrow (-1, 0)" />
   </div>
   
   The coordinates of the rotation result vertex are <InlineMath math="(-1, 0)" />.

   <LineEquation
     title={<>Rotation of Function <InlineMath math="f(x) = x^2 - 2x + 1" /></>}
     description="180° rotation around origin."
     cameraPosition={[10, 6, 10]}
     data={[
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = (i - 10) * 0.3;
           const y = x * x - 2 * x + 1;
           return { x, y, z: 0 };
         }),
         color: getColor("TEAL"),
         showPoints: false,
         labels: [
           {
             text: "f(x) = x² - 2x + 1",
             at: 18,
             offset: [1, 0.5, 0],
           },
         ],
       },
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = (i - 10) * 0.3;
           const y = x * x - 2 * x + 1;
           // 180° rotation: (x,y) → (-x,-y)
           const xRotated = -x;
           const yRotated = -y;
           return { x: xRotated, y: yRotated, z: 0 };
         }),
         color: getColor("VIOLET"),
         showPoints: false,
         labels: [
           {
             text: "180° Rotation Result",
             at: 8,
             offset: [-0.5, -0.5, 0],
           },
         ],
       },
     ]}
   />

3. Take two points on the line and rotate 270°:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="(x, y) \rightarrow (y, -x)" />
   <BlockMath math="(0, -2) \rightarrow (-2, 0)" />
   <BlockMath math="(1, 1) \rightarrow (1, -1)" />
   <BlockMath math="m = \frac{-1 - 0}{1 - (-2)} = \frac{-1}{3}" />
   <BlockMath math="y = -\frac{1}{3}x - \frac{2}{3}" />
   </div>

   So the equation of the rotation result line is <InlineMath math="y = -\frac{1}{3}x - \frac{2}{3}" />.

   <LineEquation
     title={<>Rotation of Line <InlineMath math="y = 3x - 2" /></>}
     description="270° rotation around origin."
     cameraPosition={[8, 6, 8]}
     data={[
       {
         points: Array.from({ length: 11 }, (_, i) => {
           const x = (i - 5) * 0.8;
           const y = 3 * x - 2;
           return { x, y, z: 0 };
         }),
         color: getColor("EMERALD"),
         showPoints: false,
         labels: [
           {
             text: "y = 3x - 2",
             at: 6,
             offset: [2, 0.5, 0],
           },
         ],
       },
       {
         points: Array.from({ length: 11 }, (_, i) => {
           const x = (i - 5) * 0.8;
           const y = 3 * x - 2;
           // 270° rotation: (x,y) → (y,-x)
           const xRotated = y;
           const yRotated = -x;
           return { x: xRotated, y: yRotated, z: 0 };
         }),
         color: getColor("AMBER"),
         showPoints: false,
         labels: [
           {
             text: "y = -⅓x - ⅔",
             at: 5,
             offset: [-0.5, 1.5, 0],
           },
         ],
       },
     ]}
   />

4. Using the rotation formula around a point with 60° angle:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="x' = (x-a) \cos \theta - (y-b) \sin \theta + a" />
   <BlockMath math="y' = (x-a) \sin \theta + (y-b) \cos \theta + b" />
   <BlockMath math="x' = (2-1) \cdot \frac{1}{2} - (5-1) \cdot \frac{\sqrt{3}}{2} + 1" />
   <BlockMath math="x' = \frac{3}{2} - 2\sqrt{3}" />
   <BlockMath math="y' = (2-1) \cdot \frac{\sqrt{3}}{2} + (5-1) \cdot \frac{1}{2} + 1" />
   <BlockMath math="y' = 3 + \frac{\sqrt{3}}{2}" />
   </div>
   
   Rotation result coordinates: <InlineMath math="(\frac{3}{2} - 2\sqrt{3}, 3 + \frac{\sqrt{3}}{2})" />

   <LineEquation
     title="Rotation of Point (2, 5) Around Point (1, 1)"
     description="60° counterclockwise rotation."
     cameraPosition={[8, 6, 8]}
     data={[
       {
         points: [
           { x: 1, y: 1, z: 0 },
           { x: 2, y: 5, z: 0 }
         ],
         color: getColor("CYAN"),
         showPoints: false,
         labels: [
           {
             text: "(2, 5)",
             at: 1,
             offset: [0.3, 0.3, 0],
           },
           {
             text: "Center (1, 1)",
             at: 0,
             offset: [0.3, -0.3, 0],
           },
         ],
         cone: { position: "end", size: 0.3 }
       },
       {
         points: [
           { x: 1, y: 1, z: 0 },
           { x: 1.5 - 2 * Math.sqrt(3), y: 3 + Math.sqrt(3) / 2, z: 0 }
         ],
         color: getColor("PINK"),
         showPoints: false,
         labels: [
           {
             text: "Rotation Result",
             at: 1,
             offset: [-0.5, 0.3, 0],
           },
         ],
         cone: { position: "end", size: 0.3 }
       },
     ]}
   />

5. Function <InlineMath math="y = \sqrt{x}" /> rotated 90° becomes:
   
   <div className="flex flex-col gap-4">
   <BlockMath math="x = -\sqrt{y}" />
   <BlockMath math="y = x^2 \text{ for } x \leq 0" />
   </div>
   
   The rotation result graph forms a parabola that opens upward with domain <InlineMath math="x \leq 0" /> and range <InlineMath math="y \geq 0" />. This is a reflection of parabola <InlineMath math="y = x^2" /> across the y-axis.

   <LineEquation
     title={<>Rotation of Function <InlineMath math="y = \sqrt{x}" /></>}
     description="90° counterclockwise rotation around origin."
     cameraPosition={[8, 6, 8]}
     data={[
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           const y = Math.sqrt(x);
           return { x, y, z: 0 };
         }),
         color: getColor("LIME"),
         showPoints: false,
         labels: [
           {
             text: "y = √x",
             at: 15,
             offset: [0.5, 0.3, 0],
           },
         ],
       },
       {
         points: Array.from({ length: 21 }, (_, i) => {
           const x = i * 0.25;
           const y = Math.sqrt(x);
           // 90° rotation: (x,y) → (-y,x)
           const xRotated = -y;
           const yRotated = x;
           return { x: xRotated, y: yRotated, z: 0 };
         }),
         color: getColor("ROSE"),
         showPoints: false,
         labels: [
           {
             text: "y = x² (x ≤ 0)",
             at: 15,
             offset: [-0.8, 0.3, 0],
           },
         ],
       },
     ]}
   />