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12.14 Transformations on the plane using algorithms

Worksheet
Transformations on the plane using algorithms
1

Calculate the coordinates of the newly translated points for each of the following:

a

A set of three points are given in the matrix \begin{bmatrix} -3 & -1 & -7\\-4 & -8 & -2 \end{bmatrix}. They are all translated using the transformation matrix \begin{bmatrix} 0 & 0 & 0 \\7& 7 & 7 \end{bmatrix}.

b

A set of three points are given in the matrix \begin{bmatrix} 2 & 7 & 5\\9 & 6 & 4 \end{bmatrix}. They are all translated using the transformation matrix \begin{bmatrix} 0 & 0 & 0 \\-6& -6 & -6 \end{bmatrix}.

2

Consider the following triangle:

a

Represent the points of the triangle in an array.

b

Translate each point of the triangle 7 units left and plot its graph on a coordinate plane.

c

Represent the points of the new triangle in an array.

d

Now, determine the transformation matrix.

-2
-1
1
2
3
4
5
6
x
-2
-1
1
2
3
4
5
6
7
y
3

Consider the following points:

A \left( - 2 , 1\right), B \left( - 7 , 1\right), C \left( - 7 , 3\right), D \left( - 2 , 3\right)
a

Plot the points on a coordinate plane.

b

State the coordinates of the four points if each point is translated 10 units right.

c

Now, determine the transformation matrix.

4

Determine the transformation matrix for the following:

a

Two points will be translated 5 units left.

b

Three points will be translated 6 units down.

c

Three points will be translated 7 units up and 5 units right.

5

Consider the matrix of points \begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}.

a

Plot the points on a coordinate plane.

b

Reflect the pair of points over the y-axis and write the new points in a matrix array.

c

Given three points A \left(9, 4\right), B \left(0, 6\right), C \left(3, 4\right), reflect these over the y-axis and state the coordinates of the points of the new image.

d

Consider the following algorithm for multiplying two matrices:

\begin{bmatrix} a & b \\c & d \end{bmatrix} \times \begin{bmatrix} A & B & C \\X & Y & Z \end{bmatrix}=\begin{bmatrix} aA+bX & aB+bY & aC+bZ \\cA+dX & cB+dY & cC+dZ \end{bmatrix}

Use the algorithm for multiplying two matrices to calculate the following:\begin{bmatrix} -1 & 0 \\0 & 1 \end{bmatrix} \times \begin{bmatrix} 9 & 0 & 3 \\4 & 6 & 4 \end{bmatrix}

e

Now, write down the transformation matrix.

6

Consider the matrix of points \begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}.

a

Reflect this pair of points over the x-axis and write the new points in a matrix array.

b

Given three points A \left(10, 4\right), B \left(8, 3\right), C \left(4, 4\right), reflect these over the x-axis and state the coordinates of the points of the new image.

c

Calculate: \begin{bmatrix} 1 & 0 \\0 & -1 \end{bmatrix} \times \begin{bmatrix} 10 & 8 & 4 \\4 & 3 & 4 \end{bmatrix}

d

Now, write down the transformation matrix.

7

Consider the matrix of points \begin{bmatrix} 1 & 0 \\0 & 1 \end{bmatrix}.

a

Reflect this pair of points over the line y = x and write the new points in a matrix array.

b

Given three points A \left(10, 2\right), B \left(6, 6\right), C \left(6, 0\right), reflect these over the line y = x and state the coordinates of the points of the new image.

c

Calculate: \begin{bmatrix} 0 & 1 \\1 & 0 \end{bmatrix} \times \begin{bmatrix} 10 & 6 & 6 \\2 & 6 & 0 \end{bmatrix}

d

Now, write down the transformation matrix.

8

Consider the following points:

A \left(0, 0\right), B \left(5, 0\right),C \left(5, 3\right), D \left(0, 3\right)
a

Plot the points on a coordinate plane.

b

Use the transformation matrix \begin{bmatrix} 3 & 0 \\0 & 1 \end{bmatrix} to transform the original points to its image.

c

Describe the transformation that has taken place.

9

Consider the following points:

A \left(0, 0\right), B \left(4, 0\right), C \left(4, 2\right), D \left(0, 2\right)
a

Plot the points on a coordinate plane.

b

Use the transformation matrix \begin{bmatrix} 1 & 0 \\0 & 3\end{bmatrix} to transform the original points to its image.

c

Describe the transformation that has taken place.

10

Consider the following points:

A \left(0, 0\right),B \left(3, 0\right),C \left(3, 4\right), D \left(0, 4\right)
a

Plot the points on a coordinate plane.

b

Use the transformation matrix \begin{bmatrix} 0.5 & 0 \\0 & 0.5\end{bmatrix} to transform the original points to its image.

c

Describe the transformation that has taken place.

11

Consider the points E \left(1, 0\right) and F \left(0, 1\right) in the given graph:

a

Rotate the points 90 \degree clockwise, and write the resulting points in a matrix.

b

Given the points A\left(1,-7\right), B\left(8,6\right), C\left(-2,-4\right), use the transformation matrix in part (a) to rotate the points of the image 90 \degree. Write the rotated points in a matrix.

-2
-1
1
2
3
x
-2
-1
1
2
3
y
12

Consider the points E \left(1, 0\right) and F \left(0, 1\right) in the given graph:

a

Rotate the points 180 \degree, and write the resulting points in a matrix.

b

Given the points A\left(0,-4\right), B\left(6,0\right), C\left(1,8\right), use the transformation matrix in part (a) to rotate the points of the image 180 \degree. Write the rotated points in a matrix.

-2
-1
1
2
3
x
-2
-1
1
2
3
y
13

Consider the points E \left(1, 0\right) and F \left(0, 1\right) in the given graph:

a

Rotate the points 90 \degree counterclockwise, and write the resulting points in a matrix.

b

Given the points A\left(6,-8\right), B\left(-7,7\right), C\left(-3,3\right), use the transformation matrix in part (a) to rotate the points of the image 90 \degree counterclockwise. Write the rotated points in a matrix.

-2
-1
1
2
3
x
-2
-1
1
2
3
y
14

Consider the two points with array\begin{bmatrix} 1 & 0 \\0 & 1\end{bmatrix}.

a

Determine the transformation matrix that reflects points over the x-axis.

b

Determine the transformation matrix that will rotate the points 90 \degree counterclockwise.

c

Consider the points A, B, C in the given graph.

Use the transformation matrix from part (a) to reflect the points A, B, C about the x-axis. Write the new points, A', B', C', in a matrix array.

-9
-8
-7
-6
-5
-4
-3
-2
-1
1
x
-1
1
2
3
4
5
6
7
8
9
y
d

Use the transformation matrix from part (b) to rotate the points A', B', C'. Write the new points in a matrix array.

15

Consider the two points with array\begin{bmatrix} 1 & 0 \\0 & 1\end{bmatrix}.

a

Determine the transformation matrix that reflects the points over the line y = x.

b

Determine the transformation matrix that will rotate points 90 \degree clockwise.

c

Consider the points A, B, C in the given graph:

Use the transformation matrix from part (a) to reflect the points A, B, C. Write the new points, A', B', C', in a matrix array.

-2
2
4
6
8
x
-2
2
4
6
8
y
d

Use the transformation matrix from part (b) to rotate the points A', B', C'. Write the new points in a matrix array.

16

Consider the two points with array: \begin{bmatrix} 1 & 0 \\0 & 1\end{bmatrix}.

a

Determine the transformation matrix that reflects points over the y-axis.

b

Determine the transformation matrix that will dilate points by scale factor 3 in all directions.

c

Consider the points A, B, C in the given graph:

Use the transformation matrix from part (a) to reflect the points A, B, C. Write the new points, A', B', C', in a matrix array.

-2
2
4
6
8
10
x
-10
-8
-6
-4
-2
2
y
d

Use the transformation matrix from part (b) to dilate the points A', B', C'. Write the new points in a matrix array.

17

Consider the two points with array\begin{bmatrix} 1 & 0 \\0 & 1\end{bmatrix}.

a

Determine the transformation matrix that rotates points 180 \degree.

b

Determine the transformation matrix that will reflect the points around the line y = - x.

c

Consider the points A, B, C in the given graph:

Use the transformation matrix from part (a) to rotate the points A, B, C. Write the new points, A', B', C', in a matrix array.

-2
2
4
6
8
10
x
-2
2
4
6
8
10
y
d

Use the transformation matrix from part (b) to reflect the points A', B', C'. Write the new points in a matrix array.

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VCMNA334

Implement algorithms using data structures in a general-purpose programming language

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