**How to Rotate Graphs**

**Introduction**

The key is to use parametric equations in our rotation. Using the rotation angle θ, the rotation
matrix is:

R = [ [ cos θ, -sin θ ] [ sin θ, cos θ ] ]

With the equations x(t), y(t) set as the matrix:

M = [ [ x(t) ] [ y(t) ] ]

The rotated graph is:

[ [ x’(t) ] [ y’(t) ] ] = R * M

Where:

x’(t) = x(t) * cos θ – y(t) * sin θ

y’(t) = x(t) * sin θ + y(t) * cos θ

Rotation the Function y = f(x)

Let x = t and set the parametric functions:

x(t) = t

y(t) = f(t)

Rotating the Polar Equation r = f(t) (where t = θ)

1. Solve for t.

2. Substitute r and t
in the following equations:

x(t) = r * cos t

y(t) = r * sin t

3. Simplify as
needed.

Some trigonometric properties:

sin^2 ϕ + cos^2 ϕ = 1

sin(2*ϕ) = 2 * cos ϕ * sin ϕ

cos(2*ϕ) = 2 * cos^2 ϕ – 1

sin(acos ϕ) = cos(asin ϕ) = √(1
– ϕ^2)

Please refer to this link for additional details: http://edspi31415.blogspot.com/2013/01/converting-polar-equations-to.html

**Rotation Matrices for Certain Angles**

Angle 30°, π/6: R = [
[ √3/2, -1/2 ] [ 1/2, √3/2 ] ]

Angle 45°, π/4: R = [
[ √2/2, -√2/2 ] [ √2/2, √2/2 ] ]

Angle 60°, π/3: R = [
[ 1/2, -√3/2 ] [ √3/2, 1/2 ] ]

Angle 90°, π/2: R = [
[ 0, -1 ] [ 1, 0 ] ]

Angle 120°, 2π/3: R =
[ [ -1/2, -√3/2 ] [ √3/2, -1/2 ] ]

Angle 135°, 3π/4: R =
[ [ -√2/2, -√2/2 ] [ √2/2, -√2/2 ] ]

Angle 150°, 5π/6: R =
[ [ -√3/2, -1/2 ] [ 1/2, -√3/2 ] ]

Angle 180°, π: R = [
[ -1, 0 ] [ 0, -1 ] ]

Angle 210°, 7π/6: R =
[ [ -√3/2, 1/2 ] [ -1/2, -√3/2 ] ]

Angle 225°, 5π/4: R =
[ [ -√2/2, √2/2 ] [ -√2/2, -√2/2 ] ]

Angle 240°, 4π/3: R =
[ [ -1/2, √3/2 ] [ -√3/2, -1/2 ] ]

Angle 270°, 3π/2: R =
[ [ 0, 1 ] [ -1, 0 ] ]

Angle 300°, 5π/3: R =
[ [ 1/2, √3/2 ] [ -√3/2, 1/2 ] ]

Angle 315°, 7π/4: R =
[ [ √2/2, √2/2 ] [ -√2/2, √2/2 ] ]

Angle 330°, 11π/6: R
= [ [ √3/2, 1/2 ] [ -1/2, √3/2 ] ]

**Examples**

Each example is followed by a graph of the original equation
(blue) and the rotated equations (red).
I used a Casio fx-CG 50 for the screen shots.

Example 1: y = 3*x^2,
rotate 90°

We have a function in the form of y = f(x). Let’s transfer the function to parametric
form, first by assigning x = t and y = 3*t^2.
Angle mode is in radians.

With 90°, the rotation matrix is: R = [ [ 0, -1 ] [ 1, 0 ] ]

The transformed equations are:

[ [ 0, -1 ] [ 1, 0 ] ] * [ [ t ] [ 3*t^2 ] ] = [ [ -3*t^2 ]
[ t ] ]

Rotated Equations:
x’(t) = -3*t^2, y’(t) = t

Example 2: x = t^3, y
= 2*t – 1, rotate 270°

We have the equations in parametric form. We’ll need the rotation matrix, where:

R = [ [ 0, 1 ] [ -1,
0 ] ]

[ [ 0, 1 ] [ -1, 0 ]
] * [ [ t^3 ] [ 2*t – 1 ] ] = [ [ 2*t – 1 ] [ -t^3 ] ]

Rotated Equations: x’(t)
= 2*t – 1, y’(t) = -t^3

Example 3: x = sin t,
y = e^t, rotate 135°

The rotation matrix is R = [ [ -√2/2, -√2/2 ] [ √2/2, -√2/2
] ]

Rotated Equations:
x’(t) = -√2/2 * (sin t + e^t), y’(t) = √2/2 * (sin t – e^t)

Example 4: r = 2 θ,
rotate 60°

The parametric form is x(t) = 2 * t * cos t, y(t) = 2 * t *
sin t

The rotation matrix is R = [ [ 1/2, -√3/2 ] [ √3/2, 1/2 ] ]

Rotated Equations: x’(t) = t * cos t - √3 * t * sin t, y’(t)
= t * sin t + √3 * t * cos t

Eddie

This blog is property of Edward Shore, 2017

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