Rotation

Turning a Graph around a Fixed Point

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 a wheel turning. Each point on the wheel moves along a circle centered at the axle. A graph rotation works the same way. Each point moves around the rotation center while keeping the same distance from it.

Rotation Formula Around Center Point

To rotate point $(x, y)$ around center $(a, b)$ through angle $\theta$ , first find the point's coordinates relative to the center. Call these coordinates $u$ and $v$ :

\begin{aligned} u &= x-a \\ v &= y-b \end{aligned}

Rotate these relative coordinates, then add the center coordinates back. Adding $a$ to the horizontal coordinate and $b$ to the vertical coordinate places the rotated point in the original coordinate system:

\begin{aligned} x' &= u\cos\theta-v\sin\theta+a \\ y' &= u\sin\theta+v\cos\theta+b \end{aligned}

The symbols in these equations mean:

$(x, y)$ 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

Rotation around the origin $(0, 0)$ is the case students meet first, because substituting $a = 0$ and $b = 0$ removes the translation terms from the general formula. What remains is a pure change of direction, and the three quarter turns below cover most textbook problems.

Each step is a quarter turn counterclockwise, so applying the rule twice gives the half turn and applying it three times gives the three quarter turn.

$90^\circ$ Rotation

$(x, y) \rightarrow (-y, x)$
$180^\circ$ Rotation

$(x, y) \rightarrow (-x, -y)$
$270^\circ$ Rotation

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

Visualization of Quadratic Function Rotation

Rotating a curve means rotating every individual point. The quadratic function $y = x^2$ is a convenient choice for that check because its points come in pairs with the same value, so you can verify the rotation by hand. The display below shows the original parabola and its $90^\circ$ counterclockwise rotation around the origin.

Rotation of Function

y = x^2

Around Origin

Comparison of original function with

90^\circ

counterclockwise rotation result.

Properties of Function Rotation

Rotation moves a graph around a fixed point without changing its size, so the graph keeps its shape and only its orientation changes. The table below lists what stays fixed and what changes.

Property	What happens	Why it holds
Shape	The shape of the graph does not change	Rotation moves every point by the same angle around the same center
Distances	The distance between two points stays the same	Rotating two points around one center preserves the segment that joins them
Angles between sides	Angles in the figure stay the same	A rotation is a rigid motion, so it never bends the figure
Orientation	The graph turns toward a new direction	Every point changes its position relative to the center
Compounding	Two rotations around the same center add their angles	Turning by $\theta_1$ and then by $\theta_2$ is a single turn by $\theta_1 + \theta_2$

Because distances and angles never change, a rotation is called a rigid motion. That single fact explains why shape and size survive, while position and direction are the only things that move.

Application of Rotation to Various Functions

Rotation does not treat every curve the same way. A straight line and an exponential curve both keep their shape, but the rotated line stays a straight line while the rotated exponential curve changes which axis it approaches. The two cases below show that difference.

Linear Function Rotation

For a linear function $y = mx + c$ , every point moves to a new position, so slope and position change together. The $45^\circ$ counterclockwise rotation around the origin shows how the original line becomes a second line.

Rotation of Linear Function

y = 2x + 1

45^\circ

rotation around origin.

Exponential Function Rotation

An exponential curve can also be rotated point by point. After rotation, the curve may no longer represent a function of the horizontal coordinate.

Rotation of Function

y = 2^x

90^\circ

counterclockwise rotation.

Steps to Determine Rotation Results

When you rotate a whole curve, the rotation formula applies to points, so the work runs point by point. Start by fixing the center and the angle, because both of them appear in every calculation that follows.

Next, choose several points on the original curve, including its endpoints and any turning point, and apply the rotation formula to each one. Plot the rotated points, join them in the original order, and check a few extra points between the samples. The result may no longer be a function of $x$ , because two rotated points can share the same $x$ value.

Exercises

Every problem below asks for one rotation, so each one reduces to placing the center and angle into the rotation formula. Work point by point and keep the direction of the angle in mind, because a counterclockwise turn and a clockwise turn give different results. Try them without the answer key first, then compare your steps with the worked answers.

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

Answer Key

Every worked answer starts by naming the center and the angle, then substitutes them into the rotation formula. The final coordinates are checked against the original point to confirm that the distance to the center did not change.

Using the $90^\circ$ rotation formula:
$(x, y) \rightarrow (-y, x)$
So the rotation result is $(-4, 3)$ .

Visualization of Point $(3, 4)$ Rotation
$90^\circ$ counterclockwise rotation around origin.
Original vertex rotated $180^\circ$ :
$(x, y) \rightarrow (-x, -y)$
The coordinates of the rotation result vertex are $(-1, 0)$ .

Rotation of Function $f(x) = x^2 - 2x + 1$
$180^\circ$ rotation around origin.
Take two points on the line and rotate $270^\circ$ :
$(x, y) \rightarrow (y, -x)$
So the equation of the rotation result line is $y = -\frac{1}{3}x - \frac{2}{3}$ .

Rotation of Line $y = 3x - 2$
$270^\circ$ rotation around origin.
Find the coordinates of point $(2, 5)$ relative to center $(1, 1)$ :

$\begin{aligned} u &= 2-1=1 \\ v &= 5-1=4 \end{aligned}$

Substitute these values and the angle $60^\circ$ into the rotation formulas. Both center coordinates equal $1$ , so each formula includes an added $1$ :
$\begin{aligned} x' &= u\cos\theta-v\sin\theta+a \\ &= 1\cdot\frac{1}{2}-4\cdot\frac{\sqrt{3}}{2}+1 \\ &= \frac{3}{2}-2\sqrt{3} \end{aligned}$
Rotation result coordinates: $(\frac{3}{2} - 2\sqrt{3}, 3 + \frac{\sqrt{3}}{2})$

Rotation of Point $(2, 5)$ Around Point $(1, 1)$
$60^\circ$ counterclockwise rotation.
Function $y = \sqrt{x}$ rotated $90^\circ$ becomes:
$x = -\sqrt{y}$
The rotated curve is the left half of an upward-opening parabola. Its domain is $x \leq 0$ and its range is $y \geq 0$ . It is obtained by reflecting the original right half of $y = x^2$ across the $y$ -axis.

Rotation of Function $y = \sqrt{x}$
$90^\circ$ counterclockwise rotation around origin.

Turning a Graph around a Fixed Point

Rotation Formula Around Center Point

To rotate point $(x, y)$ around center $(a, b)$ through angle $\theta$ , first find the point's coordinates relative to the center. Call these coordinates $u$ and $v$ :

\begin{aligned} u &= x-a \\ v &= y-b \end{aligned}

\begin{aligned} x' &= u\cos\theta-v\sin\theta+a \\ y' &= u\sin\theta+v\cos\theta+b \end{aligned}

The symbols in these equations mean:

$(x, y)$ 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

Each step is a quarter turn counterclockwise, so applying the rule twice gives the half turn and applying it three times gives the three quarter turn.

$90^\circ$ Rotation

$(x, y) \rightarrow (-y, x)$
$180^\circ$ Rotation

$(x, y) \rightarrow (-x, -y)$
$270^\circ$ Rotation

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

Visualization of Quadratic Function Rotation

Rotation of Function

y = x^2

Around Origin

Comparison of original function with

90^\circ

counterclockwise rotation result.

Properties of Function Rotation

Rotation moves a graph around a fixed point without changing its size, so the graph keeps its shape and only its orientation changes. The table below lists what stays fixed and what changes.

Property	What happens	Why it holds
Shape	The shape of the graph does not change	Rotation moves every point by the same angle around the same center
Distances	The distance between two points stays the same	Rotating two points around one center preserves the segment that joins them
Angles between sides	Angles in the figure stay the same	A rotation is a rigid motion, so it never bends the figure
Orientation	The graph turns toward a new direction	Every point changes its position relative to the center
Compounding	Two rotations around the same center add their angles	Turning by $\theta_1$ and then by $\theta_2$ is a single turn by $\theta_1 + \theta_2$

Because distances and angles never change, a rotation is called a rigid motion. That single fact explains why shape and size survive, while position and direction are the only things that move.

Application of Rotation to Various Functions

Linear Function Rotation

Rotation of Linear Function

y = 2x + 1

45^\circ

rotation around origin.

Exponential Function Rotation

An exponential curve can also be rotated point by point. After rotation, the curve may no longer represent a function of the horizontal coordinate.

Rotation of Function

y = 2^x

90^\circ

counterclockwise rotation.

Steps to Determine Rotation Results

Exercises

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

Answer Key

Using the $90^\circ$ rotation formula:
$(x, y) \rightarrow (-y, x)$
So the rotation result is $(-4, 3)$ .

Visualization of Point $(3, 4)$ Rotation
$90^\circ$ counterclockwise rotation around origin.
Original vertex rotated $180^\circ$ :
$(x, y) \rightarrow (-x, -y)$
The coordinates of the rotation result vertex are $(-1, 0)$ .

Rotation of Function $f(x) = x^2 - 2x + 1$
$180^\circ$ rotation around origin.
Take two points on the line and rotate $270^\circ$ :
$(x, y) \rightarrow (y, -x)$
So the equation of the rotation result line is $y = -\frac{1}{3}x - \frac{2}{3}$ .

Rotation of Line $y = 3x - 2$
$270^\circ$ rotation around origin.
Find the coordinates of point $(2, 5)$ relative to center $(1, 1)$ :

$\begin{aligned} u &= 2-1=1 \\ v &= 5-1=4 \end{aligned}$

Substitute these values and the angle $60^\circ$ into the rotation formulas. Both center coordinates equal $1$ , so each formula includes an added $1$ :
$\begin{aligned} x' &= u\cos\theta-v\sin\theta+a \\ &= 1\cdot\frac{1}{2}-4\cdot\frac{\sqrt{3}}{2}+1 \\ &= \frac{3}{2}-2\sqrt{3} \end{aligned}$
Rotation result coordinates: $(\frac{3}{2} - 2\sqrt{3}, 3 + \frac{\sqrt{3}}{2})$

Rotation of Point $(2, 5)$ Around Point $(1, 1)$
$60^\circ$ counterclockwise rotation.
Function $y = \sqrt{x}$ rotated $90^\circ$ becomes:
$x = -\sqrt{y}$
The rotated curve is the left half of an upward-opening parabola. Its domain is $x \leq 0$ and its range is $y \geq 0$ . It is obtained by reflecting the original right half of $y = x^2$ across the $y$ -axis.

Rotation of Function $y = \sqrt{x}$
$90^\circ$ counterclockwise rotation around origin.