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Transformation of Graphs Using Matrices - Dilation

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Transformation of Graphs Using Matrices - Dilation

Study Guide

Key Definition

Dilation involves changing the size of a shape using a center point and a scale factor. For dilations centered at the origin, this is represented as $k \times \begin{bmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{bmatrix}$. For a dilation with center $(h,k')$, translate the shape by $(-h,-k')$, apply the origin-centered dilation, then translate back by $(h,k')$.

Important Notes

Dilation requires a $scale\ factor$.
The scale factor $k$ determines the degree of enlargement or reduction.
Center of dilation is crucial for calculations; for non-origin centers, translate the shape so the center moves to the origin, apply the scale factor, then translate back.
Each coordinate is multiplied by the scale factor.
Dilation affects the size but not the shape.

Mathematical Notation

$\times$ represents multiplication

$\begin{bmatrix} x \\ y \end{bmatrix}$ is a point in matrix form

$k$ is the scale factor

$\begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$ is the dilation matrix for scale factor k

Remember to use proper notation when solving problems

Why It Works

Dilation using matrices allows each point to be scaled uniformly by multiplying each coordinate by the scale factor, $k$. For a center $C$, the formula becomes $v' = C + D(v - C)$ where $D = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$.

Remember

To dilate a shape, multiply each point by the scale factor: $k \times \begin{bmatrix} x \\ y \end{bmatrix}$. For non-origin centers, translate, scale, then translate back.

Quick Reference

Dilation Formula:$k \times \begin{bmatrix} x_1 & x_2 & x_3 \\ y_1 & y_2 & y_3 \end{bmatrix}$

Understanding Transformation of Graphs Using Matrices - Dilation

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Video explanation of this concept

Beginner

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Beginner Explanation

A dilation means resizing a shape by multiplying its coordinates by a scale factor. For example, to dilate the point $[x, y]$ by a scale factor $k$, compute $[k x, k y]$. If $k > 1$, the shape enlarges; if $0 < k < 1$, it shrinks.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the result of dilating point $\begin{bmatrix} 3 \\ 4 \end{bmatrix}$ by a scale factor of 2?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are creating a digital animation and need to enlarge a character by a scale factor of 3 using matrix dilation. If the character's position is $\begin{bmatrix} 2 \\ 5 \end{bmatrix}$, what will be the new position?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

What happens to a triangle's area when it is dilated by a scale factor of $k$?

Challenge Quiz

Single Choice Quiz

Advanced

Given a dilation matrix $\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}$, what is the result of applying it to a point $\begin{bmatrix} -1 \\ 3 \end{bmatrix}$?

Recap

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