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Precalculus with Trigonometry: Concepts and Applications

Paul A. Foerster

Chapter 11

Matrix Transformations and Fractal Figures - all with Video Answers

Educators

Section 1

Introduction to Iterated Transformations

03:22

Problem 1

The left diagram in Figure $11-1$ a shows a 10 -cm by 10 -cm square. To create the middle diagram, the original, or pre-image, square was transformed into four similar squares, each with sides that are $40 \%$ of the original side length. These image squares were then translated so that each has a corner at
one of the corners of the pre-image. The right diagram shows the result of applying the same transformation to each of the four squares from the first iteration. In this problem set you will explore the perimeter and area of various iterations.
Find the perimeter and area of the pre-image square. Find the total perimeter and area of the four squares in the first iteration. Find the total perimeter and area of the 16 squares in the second iteration. Display the answers in a table with these column headings: Iteration number, Side length, Total perimeter, and Total area.
(FIGURE CAN'T COPY)

Wendi Zhao
Wendi Zhao
Numerade Educator
02:17

Problem 2

The left diagram in Figure $11-1$ a shows a 10 -cm by 10 -cm square. To create the middle diagram, the original, or pre-image, square was transformed into four similar squares, each with sides that are $40 \%$ of the original side length. These image squares were then translated so that each has a corner at
one of the corners of the pre-image. The right diagram shows the result of applying the same transformation to each of the four squares from the first iteration. In this problem set you will explore the perimeter and area of various iterations.
What pattern do you notice that relates the total perimeter to the iteration number? What pattern relates the total area to the iteration number? What pattern relates the total area to the total perimeter?
(FIGURE CAN'T COPY)

Jay Patel
Jay Patel
Numerade Educator
03:22

Problem 3

The left diagram in Figure $11-1$ a shows a 10 -cm by 10 -cm square. To create the middle diagram, the original, or pre-image, square was transformed into four similar squares, each with sides that are $40 \%$ of the original side length. These image squares were then translated so that each has a corner at
one of the corners of the pre-image. The right diagram shows the result of applying the same transformation to each of the four squares from the first iteration. In this problem set you will explore the perimeter and area of various iterations.
Using the patterns you observed in Problem 2 find the total perimeter and total area of the third and fourth iterations.

Wendi Zhao
Wendi Zhao
Numerade Educator
02:27

Problem 4

The left diagram in Figure $11-1$ a shows a 10 -cm by 10 -cm square. To create the middle diagram, the original, or pre-image, square was transformed into four similar squares, each with sides that are $40 \%$ of the original side length. These image squares were then translated so that each has a corner at
one of the corners of the pre-image. The right diagram shows the result of applying the same transformation to each of the four squares from the first iteration. In this problem set you will explore the perimeter and area of various iterations.
Calculate the total perimeter and the total area of the 20 th iteration.

Luca Alexander
Luca Alexander
Numerade Educator
02:46

Problem 5

The left diagram in Figure $11-1$ a shows a 10 -cm by 10 -cm square. To create the middle diagram, the original, or pre-image, square was transformed into four similar squares, each with sides that are $40 \%$ of the original side length. These image squares were then translated so that each has a corner at
one of the corners of the pre-image. The right diagram shows the result of applying the same transformation to each of the four squares from the first iteration. In this problem set you will explore the perimeter and area of various iterations.
If the iterations could be carried on infinitely many times, the images would approach a figure called Sierpinski's carpet or Sierpinski's square. What would be the total perimeter of this figure? What would be the total area? Does the answer surprise you? (In this chapter you will encounter other surprises, such as the fact that this figure is less than two-dimensional but more than one-dimensional!)

Aman Gupta
Aman Gupta
Numerade Educator