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Transformation of Graphs Using Matrices - <a href="transformation-of-graphs-using-matrices-translation">Translation</a>

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Transformation of Graphs Using Matrices - <a href="transformation-of-graphs-using-matrices-translation">Translation</a>

Study Guide

Key Definition

A translation moves a shape without rotating or resizing it, represented by $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x + a \\ y + b \end{pmatrix}$

Important Notes

Translation does not affect the size or shape of the graph.
The matrix for translation is $\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$ for 2D transformations.
Horizontal translations affect the $x$-values.
Vertical translations affect the $y$-values.
Translation matrices can be combined with other transformation matrices.

Mathematical Notation

$\begin{pmatrix} x' \\ y' \end{pmatrix}$ represents the new coordinates after translation

$+$ represents addition

$-$ represents subtraction

$(x, y) + (a, b) = (x + a, y + b)$ represents vector translation

$\begin{pmatrix}1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$ is the homogeneous translation matrix

$\begin{pmatrix} x \\ y \\ 1\end{pmatrix}$ is a point in homogeneous coordinates

Remember to use proper notation when solving problems

Why It Works

Translation matrices allow coordinate points to shift by adding constants to the original coordinates: $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x + a \\ y + b \end{pmatrix}$

Remember

For translation, use $\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} x + a \\ y + b \end{pmatrix}$

Quick Reference

Translation Matrix:$\begin{pmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$

Understanding Transformation of Graphs Using Matrices - <a href="transformation-of-graphs-using-matrices-translation">Translation</a>

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A translation shifts every point of a graph by adding a constant vector (a, b). For any point (x, y), the new position is (x + a, y + b). This moves the graph horizontally by a units and vertically by b units without changing its shape.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the new position of a point $(x, y)$ after translating 3 units right and 4 units up?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

You are designing a game where a character moves $5$ units to the left and $2$ units up after a power-up.

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to the triangle vertices $T(2, -1)$, $R(4, 3)$, $I(-3, -2)$ when translated $5$ units left and $2$ units up?

Challenge Quiz

Single Choice Quiz

Advanced

Determine the effect of a translation matrix $\begin{pmatrix} 1 & 0 & -6 \\ 0 & 1 & 8 \\ 0 & 0 & 1 \end{pmatrix}$ on point $(x, y)$.

Recap

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Review key concepts and takeaways