Master

Transformation of Graphs Using Matrices - Rotations

Master transformation of graphs using matrices - rotations with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Transformation of Graphs Using Matrices - Rotations

Study Guide

Key Definition

A rotation is a transformation that turns every point of a preimage around a fixed point called the center of rotation by a specified angle and direction.

Important Notes

Rotations can be clockwise or counterclockwise
The center of rotation is often the origin $(0, 0)$, but other centers are possible
The amount of rotation is measured in degrees, called the angle of rotation
Rotation matrices are used to perform rotations algebraically
The standard rotation matrix for a $90^\circ$ counterclockwise rotation is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
The general rotation matrix for a counterclockwise rotation by an angle $\theta$ is $\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$

Mathematical Notation

$\times$ represents multiplication

$\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$ are trigonometric functions

$\frac{a}{b}$ represents a fraction

$\angle$ represents an angle

Remember to use proper notation when solving problems

Why It Works

Rotation matrices transform the coordinates of the graph by maintaining the distance from the center and changing the orientation according to the angle.

Remember

The rotation matrix for a counterclockwise rotation by an angle $\theta$ is $\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$. For example, for $90^\circ$, it is $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.

Quick Reference

General Counterclockwise Rotation:$\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{pmatrix}$

90° Counterclockwise:$\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

180° Rotation:$\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$

270° Counterclockwise:$\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

Understanding Transformation of Graphs Using Matrices - Rotations

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A rotation of a point in the plane can be performed using a rotation matrix. For example, a counterclockwise rotation of 90° uses $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Multiplying this by (x, y) gives the rotated point (-y, x). For instance, rotating (1, 0) yields (0, 1).

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the result of rotating the point $(1, 0)$ by $90^\circ$ counterclockwise around the origin?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

A skateboarder rides up a ramp, rotating their board $180^\circ$ in the air. How can we represent this rotation using a matrix?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a point $(x, y)$ is rotated counterclockwise by $90^\circ$, what is the new position?

Challenge Quiz

Single Choice Quiz

Advanced

What matrix represents a $270^\circ$ counterclockwise rotation?

Recap

Watch & Learn

Review key concepts and takeaways