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Transformation of Graphs Using Matrices - Reflection

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Transformation of Graphs Using Matrices - Reflection

Study Guide

Key Definition

A reflection is a transformation that 'flips' an image over a line of symmetry. Reflection matrices include: $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ for the x-axis, $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$ for the y-axis, and $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ for the line $y = x$.

Important Notes

Reflection preserves size and shape but reverses orientation of the figure.
Reflection over x-axis converts $f(x)$ to $-f(x)$.
Reflection over y-axis converts $f(x)$ to $f(-x)$.
Reflection uses matrices to map each point across a line of symmetry.
Matrix multiplication for reflection is performed by placing the reflection matrix on the left side.

Mathematical Notation

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ represents a 2x2 matrix

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Reflections using matrices work by applying the matrix to each point's coordinates, effectively flipping points across the desired axis or line using $matrix \, multiplication$.

Remember

To reflect a point across an axis or line, use the corresponding reflection matrix with $correct \; setup$.

Quick Reference

Reflection over x-axis:$\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$

Reflection over y-axis:$\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

Reflection over line y = x:$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$

Understanding Transformation of Graphs Using Matrices - Reflection

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A reflection flips an image across a line using matrices like $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which matrix reflects over the y-axis?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a logo that needs to be symmetric. Given points (1, 2), (3, -1), and (-2, 4), use the reflection matrix $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ to reflect these points accurately over the x-axis.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Explore how the reflection matrix $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ affects the points (1, 2), (0, -3), and (-2, 1).

Challenge Quiz

Single Choice Quiz

Advanced

Choose the correct matrix for reflecting over the line $y = x$.

Recap

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Review key concepts and takeaways