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URL: https://nakafa.com/en/subjects/mathematics/function-transformation/rotation
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/function-transformation/rotation/en.mdx
Learn function rotation around fixed points with worked examples. Learn rotation formulas, angles, and transformations for various function types.
---
## 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 $$(a, b)$$ with angle $$\theta$$, the transformation formula is:
Visible text: For rotation around center point with angle , the transformation formula is:
Component: MathContainer
Children:
```math
x' = (x - a) \cos \theta - (y - b) \sin \theta + a
```
```math
y' = (x - a) \sin \theta + (y - b) \cos \theta + b
```
Where:
- $$(x, y)$$ is the original point coordinate
- is the rotated point coordinate
- $$(a, b)$$ is the rotation center point
- $$\theta$$ is the rotation angle (positive for counterclockwise direction)
Visible text: - is the original point coordinate
- is the rotated point coordinate
- is the rotation center point
- is the rotation angle (positive for counterclockwise direction)
## Special Rotation Around Origin
When rotation is performed around the origin $$(0, 0)$$, the formula becomes simpler:
Visible text: When rotation is performed around the origin , the formula becomes simpler:
1. $$90^\circ$$ Rotation
```math
(x, y) \rightarrow (-y, x)
```
2. $$180^\circ$$ Rotation
```math
(x, y) \rightarrow (-x, -y)
```
3. $$270^\circ$$ Rotation
```math
(x, y) \rightarrow (y, -x)
```
Visible text: 1. Rotation
2. Rotation
3. Rotation
## Visualization of Quadratic Function Rotation
Let's see how rotation affects the quadratic function $$y = x^2$$:
Visible text: Let's see how rotation affects the quadratic function :
Component: LineEquation
Props:
- title: Rotation of Function $$y = x^2$$ Around Origin
Visible text: Rotation of Function Around Origin
- description: Comparison of original function with $$90^\circ$$ counterclockwise rotation result.
Visible text: Comparison of original function with 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 $$y = mx + c$$, rotation will change the slope and position of the line.
Visible text: For linear function , rotation will change the slope and position of the line.
Component: LineEquation
Props:
- title: Rotation of Linear Function $$y = 2x + 1$$
Visible text: Rotation of Linear Function
- description: $$45^\circ$$ rotation around origin.
Visible text: 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.
Component: LineEquation
Props:
- title: Rotation of Function $$y = 2^x$$
Visible text: Rotation of Function
- description: $$90^\circ$$ counterclockwise rotation.
Visible text: 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 $$(3, 4)$$ around the origin with a $$90^\circ$$ counterclockwise angle.
2. Function $$f(x) = x^2 - 2x + 1$$ is rotated $$180^\circ$$ around the origin. Determine the coordinates of the rotation result vertex if the original vertex is at $$(1, 0)$$.
3. Line $$y = 3x - 2$$ is rotated $$270^\circ$$ around the origin. Determine the equation of the rotation result line.
4. Point $$(2, 5)$$ is rotated $$60^\circ$$ around point $$(1, 1)$$. Determine the rotation result coordinates.
5. Function $$y = \sqrt{x}$$ for $$x \geq 0$$ is rotated $$90^\circ$$ counterclockwise around the origin. Explain the shape of the rotation result graph.
Visible text: 1. Determine the rotation result of point around the origin with a counterclockwise angle.
2. Function is rotated around the origin. Determine the coordinates of the rotation result vertex if the original vertex is at .
3. Line is rotated around the origin. Determine the equation of the rotation result line.
4. Point is rotated around point . Determine the rotation result coordinates.
5. Function for is rotated counterclockwise around the origin. Explain the shape of the rotation result graph.
### Answer Key
1. Using the $$90^\circ$$ rotation formula:
```math
(x, y) \rightarrow (-y, x)
```
```math
(3, 4) \rightarrow (-4, 3)
```
So the rotation result is $$(-4, 3)$$.
Visualization of Point $$(3, 4)$$ Rotation}
description={<>$$90^\circ$$ 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 $$180^\circ$$:
```math
(x, y) \rightarrow (-x, -y)
```
```math
(1, 0) \rightarrow (-1, 0)
```
The coordinates of the rotation result vertex are $$(-1, 0)$$.
Rotation of Function $$f(x) = x^2 - 2x + 1$$}
description={<>$$180^\circ$$ 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 $$270^\circ$$:
```math
(x, y) \rightarrow (y, -x)
```
```math
(0, -2) \rightarrow (-2, 0)
```
```math
(1, 1) \rightarrow (1, -1)
```
```math
m = \frac{-1 - 0}{1 - (-2)} = \frac{-1}{3}
```
```math
y = -\frac{1}{3}x - \frac{2}{3}
```
So the equation of the rotation result line is $$y = -\frac{1}{3}x - \frac{2}{3}$$.
Rotation of Line $$y = 3x - 2$$}
description={<>$$270^\circ$$ 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 $$60^\circ$$ angle:
Rotation result coordinates: $$(\frac{3}{2} - 2\sqrt{3}, 3 + \frac{\sqrt{3}}{2})$$
Rotation of Point $$(2, 5)$$ Around Point $$(1, 1)$$}
description={<>$$60^\circ$$ 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 $$y = \sqrt{x}$$ rotated $$90^\circ$$ becomes:
```math
x = -\sqrt{y}
```
```math
y = x^2 \text{ for} x \leq 0
```
The rotation result graph forms a parabola that opens upward with domain $$x \leq 0$$ and range $$y \geq 0$$. This is a reflection of parabola $$y = x^2$$ across the $$y$$-axis.
Rotation of Function $$y = \sqrt{x}$$}
description={<>$$90^\circ$$ 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],
},
],
},
]}
/>
Visible text: 1. Using the rotation formula:
So the rotation result is .
Visualization of Point Rotation}
description={<> 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 :
The coordinates of the rotation result vertex are .
Rotation of Function }
description={<> 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 :
So the equation of the rotation result line is .
Rotation of Line }
description={<> 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 angle:
Rotation result coordinates:
Rotation of Point Around Point }
description={<> 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 rotated becomes:
The rotation result graph forms a parabola that opens upward with domain and range . This is a reflection of parabola across the -axis.
Rotation of Function }
description={<> 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],
},
],
},
]}
/>