Rotation

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:

x' = (x - a) \cos \theta - (y - b) \sin \theta + a

y' = (x - a) \sin \theta + (y - b) \cos \theta + b

Where:

$(x, y)$ is the original point coordinate
$(x', y')$ is the rotated point coordinate
$(a, b)$ is the rotation center point
$\theta$ 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:

90 Degree Rotation

$(x, y) \rightarrow (-y, x)$
180 Degree Rotation

$(x, y) \rightarrow (-x, -y)$
270 Degree Rotation

$(x, y) \rightarrow (y, -x)$

Visualization of Quadratic Function Rotation

Let's see how rotation affects the quadratic function $y = x^2$ :

Rotation of Function

y = x^2

Around Origin

Comparison of original function with 90° counterclockwise rotation result.

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.

Rotation of Linear Function

y = 2x + 1

45° rotation around origin.

Exponential Function Rotation

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

Rotation of Function

y = 2^x

90° counterclockwise rotation.

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

Determine the rotation result of point $(3, 4)$ around the origin with a 90° counterclockwise angle.
Function $f(x) = x^2 - 2x + 1$ is rotated 180° around the origin. Determine the coordinates of the rotation result vertex if the original vertex is at $(1, 0)$ .
Line $y = 3x - 2$ is rotated 270° around the origin. Determine the equation of the rotation result line.
Point $(2, 5)$ is rotated 60° around point $(1, 1)$ . Determine the rotation result coordinates.
Function $y = \sqrt{x}$ for $x \geq 0$ is rotated 90° counterclockwise around the origin. Explain the shape of the rotation result graph.

Answer Key

Using the 90° rotation formula:

$(x, y) \rightarrow (-y, x)$
$(3, 4) \rightarrow (-4, 3)$

So the rotation result is $(-4, 3)$ .

Visualization of Point (3, 4) Rotation
90° counterclockwise rotation around origin.
Original vertex rotated 180°:

$(x, y) \rightarrow (-x, -y)$
$(1, 0) \rightarrow (-1, 0)$

The coordinates of the rotation result vertex are $(-1, 0)$ .

Rotation of Function $f(x) = x^2 - 2x + 1$
180° rotation around origin.
Take two points on the line and rotate 270°:

$(x, y) \rightarrow (y, -x)$
$(0, -2) \rightarrow (-2, 0)$
$(1, 1) \rightarrow (1, -1)$
$m = \frac{-1 - 0}{1 - (-2)} = \frac{-1}{3}$
$y = -\frac{1}{3}x - \frac{2}{3}$

So the equation of the rotation result line is $y = -\frac{1}{3}x - \frac{2}{3}$ .

Rotation of Line $y = 3x - 2$
270° rotation around origin.
Using the rotation formula around a point with 60° angle:

$x' = (x-a) \cos \theta - (y-b) \sin \theta + a$
$y' = (x-a) \sin \theta + (y-b) \cos \theta + b$
$x' = (2-1) \cdot \frac{1}{2} - (5-1) \cdot \frac{\sqrt{3}}{2} + 1$
$x' = \frac{3}{2} - 2\sqrt{3}$
$y' = (2-1) \cdot \frac{\sqrt{3}}{2} + (5-1) \cdot \frac{1}{2} + 1$
$y' = 3 + \frac{\sqrt{3}}{2}$

Rotation result coordinates: $(\frac{3}{2} - 2\sqrt{3}, 3 + \frac{\sqrt{3}}{2})$

Rotation of Point (2, 5) Around Point (1, 1)
60° counterclockwise rotation.
Function $y = \sqrt{x}$ rotated 90° becomes:

$x = -\sqrt{y}$
$y = x^2 \text{ for } x \leq 0$

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 $y = \sqrt{x}$
90° counterclockwise rotation around origin.

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