NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 27
M1
PRECALCULUS AND ADVANCED TOPICS
Lesson 27: Getting a Handle on New Transformations
Classwork
Opening Exercise
Explain the geometric effect of each matrix.
a.
𝑎
[
𝑏
−𝑏
]
𝑎
b.
cos⁡(𝜃)
[
sin⁡(𝜃)
c.
𝑘
[
0
0
]
𝑘
d.
1
[
0
0
]
1
e.
0
[
0
0
]
0
f.
𝑎
[
𝑏
𝑐
]
𝑑
−sin⁡(𝜃)
]
cos⁡(𝜃)
Lesson 27:
Getting a Handle on New Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.138
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 27
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Example 1
Given the transformation [
𝑘
𝑘
0
] with 𝑘 > 0:
1
a.
Perform this transformation on the vertices of the unit square. Sketch the image, and label the vertices.
b.
Calculate the area of the image using the dimensions of the image parallelogram.
c.
Confirm the area of the image using the determinant.
d.
𝑥
Perform the transformation on [𝑦].
e.
In order for two matrices to be equivalent, each of the corresponding elements must be equivalent. Given
𝑥
5
that, if the image of this transformation is [ ], find [𝑦].
4
Lesson 27:
Getting a Handle on New Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.139
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 27
M1
PRECALCULUS AND ADVANCED TOPICS
f.
g.
1
Perform the transformation on [ ]. Write the image matrix.
1
Perform the transformation on the image again, and then repeat until the transformation has been performed
four times on the image of the preceding matrix.
Exercise
Perform the transformation [
𝑘
1
0
] with 𝑘 > 1 on the vertices of the unit square.
𝑘
a.
What are the vertices of the image?
b.
Calculate the area of the image.
c.
𝑥
𝑥
−2
If the image of the transformation on [𝑦] is [ ], find [𝑦] in terms of 𝑘.
−1
Example 2
Consider the matrix 𝐿 = [
a.
2
−1
5
]. For each real number 0 ≤ 𝑡 ≤ 1, consider the point (3 + 𝑡, 10 + 2𝑡).
3
Find point 𝐴 when 𝑡 = 0.
Lesson 27:
Getting a Handle on New Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.140
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 27
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
b.
Find point 𝐵 when 𝑡 = 1.
c.
1
Show that for 𝑡 = , (3 + 𝑡, 10 + 2𝑡) is the midpoint of ̅̅̅̅
𝐴𝐵 .
d.
Show that for each value of 𝑡, (3 + 𝑡, 10 + 2𝑡) is a point on the line through 𝐴 and 𝐵.
e.
Find 𝐿𝐴 and 𝐿𝐵.
f.
What is the equation of the line through 𝐿𝐴 and 𝐿𝐵?
g.
3+𝑡
Show that 𝐿 [
] lies on the line through 𝐿𝐴 and 𝐿𝐵.
10 + 2𝑡
2
Lesson 27:
Getting a Handle on New Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.141
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 27
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
PRECALCULUS AND ADVANCED TOPICS
Problem Set
1.
Perform the following transformation on the vertices of the unit square. Sketch the image, label the vertices, and
find the area of the image parallelogram.
1 0
a. [
]
1 1
2 0
b. [
]
2 1
3 0
c. [
]
3 1
2 0
d. [
]
2 2
3 0
e. [
]
3 3
1 0
f. [
]
1 2
1 0
g. [
]
1 3
2 0
h. [
]
2 3
3 0
i.
[
]
3 5
2.
Given [
a.
b.
c.
3.
𝑘𝑥
𝑘 0 𝑥
][ ] = [
]. Find the value of 𝑘 so that:
𝑘𝑥 + 𝑦
𝑘 1 𝑦
𝑥
3
24
[𝑦] = [ ] and the image is [ ].
−2
22
𝑥
27
18
[𝑦] = [ ] and the image is [ ].
3
21
Given [
a.
b.
4.
𝑥
𝑘𝑥
𝑘 0 𝑥
] [𝑦 ] = [
]. Find [𝑦] if the image of the transformation is the following:
𝑘𝑥
+
𝑦
𝑘 1
4
[ ]
5
−3
[ ]
2
5
[ ]
−6
𝑘𝑥
𝑘 0 𝑥
][ ] = [
]. Find the value of 𝑘 so that:
𝑥 + 𝑘𝑦
1 𝑘 𝑦
𝑥
−4
−12
[𝑦] = [ ] and the image is [
].
5
11
Given [
a.
5
b.
−15
𝑥
[𝑦] = [3
] and the image is [ 1 ].
2
−
3
9
Lesson 27:
Getting a Handle on New Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.142
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 27
M1
PRECALCULUS AND ADVANCED TOPICS
5.
Perform the following transformation on the vertices of the unit square. Sketch the image, label the vertices, and
find the area of the image parallelogram.
2 0
a. [
]
1 2
3 0
b. [
]
1 3
2 0
c. [
]
3 2
2 0
d. [
]
4 2
4 0
e. [
]
2 4
3 0
f. [
]
5 3
6.
Consider the matrix 𝐿 = [
a.
1 3
]. For each real number 0 ≤ 𝑡 ≤ 1, consider the point (3 + 2𝑡, 12 + 2𝑡).
2 4
Find the point 𝐴 when 𝑡 = 0.
b.
Find the point 𝐵 when 𝑡 = 1.
c.
Show that for 𝑡 = , (3 + 2𝑡, 12 + 2𝑡) is the midpoint of ̅̅̅̅
𝐴𝐵 .
d.
Show that for each value of 𝑡, (3 + 2𝑡, 12 + 2𝑡) is a point on the line through 𝐴 and 𝐵.
e.
Find 𝐿𝐴 and 𝐿𝐵.
f.
What is the equation of the line through 𝐿𝐴 and 𝐿𝐵?
3 + 2𝑡
Show that 𝐿 [
] lies on the line through 𝐿𝐴 and 𝐿𝐵.
12 + 2𝑡
g.
1
2
Lesson 27:
Getting a Handle on New Transformations
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from PreCal-M1-TE-1.3.0-07.2015
S.143
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.