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Transformation of Graphs Using Matrices - Translation

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Transformation of Graphs Using Matrices - Translation

Study Guide

Key Definition

Translation moves an image across a graph without changing its size, shape, or orientation. This can be represented using matrices.

Important Notes

Translating a graph involves moving every point by the same amount.
Translation can be vertical or horizontal.
$x \rightarrow x + a$ shifts the graph horizontally by $a$ units.
$y \rightarrow y + b$ shifts the graph vertically by $b$ units.
Using matrices for translation involves adding a vector to all coordinates.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ or $*$ represents multiplication

$\div$ or $/$ represents division

Remember to use proper notation when solving problems

Why It Works

Translation using matrices allows for efficient manipulation of graph coordinates by simple matrix addition.

Remember

Translation does not change the size or shape of the graph, only its position.

Quick Reference

Translation Matrix:$\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} a \\ b \end{bmatrix}$

Understanding Transformation of Graphs Using Matrices - Translation

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Video explanation of this concept

Beginner

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Beginner Explanation

Translation shifts every point by adding a fixed vector. For example, translating (x, y) by (a, b) gives $(x + a, y + b)$.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the new coordinate of point $(2, -1)$ after translating 5 units left and 2 units up?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a video game where a character needs to move 3 units right and 4 units down. How would you represent this movement using matrices?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Translate the triangle with vertices at $(2, -1)$, $(4, 3)$, and $(-3, -2)$ by 5 units left and 2 units up. What are the new coordinates?

Challenge Quiz

Single Choice Quiz

Advanced

If a function $f(x, y)$ is translated by matrix $\begin{bmatrix} -3 \\ 7 \end{bmatrix}$, what is the new position of $(6, 2)$?

Recap

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