Eureka Math Precalculus Module 1 Lesson 21 Answer Key

Engage NY Eureka Math Precalculus Module 1 Lesson 21 Answer Key

Eureka Math Precalculus Module 1 Lesson 21 Exercise Answer Key

Opening Exercise

Suppose that L1(x,y)=(2x-3y,3x+2y) and L2(x,y)=(3x+4y,-4y+3x).
Find the result of performing L1and then L2on a point (p,q). That is, find L2(L1(p,q)).
Answer:
L2(L1(p,q))=L2(2p-3q,3p+2q)
=(3(2p-3q)+4(3p+2q),-4(2p-3q)+3(3p+2q))
=(6p-9q+12p+8q,-8p+12q+9p+6q)
=(18p-q,p+18q)

Exercises

Exercise 1.
Calculate each of the following products.
a. \(\left(\begin{array}{cc}
3 & -2 \\
-1 & 4
\end{array}\right)\left(\begin{array}{l}
1 \\
5
\end{array}\right)\)
Answer:
\(\left(\begin{array}{c}
3-10 \\
-1+20
\end{array}\right)\) = \(\left(\begin{array}{c}
-7 \\
19
\end{array}\right)\)

b. \(\left(\begin{array}{ll}
3 & 3 \\
3 & 3
\end{array}\right)\left(\begin{array}{c}
4 \\
-4
\end{array}\right)\)
Answer:
\(\left(\begin{array}{ll}
3 & 3 \\
3 & 3
\end{array}\right)\left(\begin{array}{c}
4 \\
-4
\end{array}\right)\)\(\left(\begin{array}{l}
12-12 \\
12-12
\end{array}\right)\) = \(\left(\begin{array}{l}
\mathbf{0} \\
0
\end{array}\right)\)

c. \(\left(\begin{array}{ll}
2 & -4 \\
5 & -1
\end{array}\right)\left(\begin{array}{c}
3 \\
-2
\end{array}\right)\)
Answer:
\(\left(\begin{array}{cc}
2 & -4 \\
5 & -1
\end{array}\right)\left(\begin{array}{c}
3 \\
-2
\end{array}\right)\) = \(\) = \(\left(\begin{array}{l}
14 \\
17
\end{array}\right)\)

Exercise 2.
Find a value of k so that \(\left(\begin{array}{ll}
1 & 2 \\
k & 1
\end{array}\right)\left(\begin{array}{c}
3 \\
-1
\end{array}\right)\)=\(\left(\begin{array}{c}
1 \\
11
\end{array}\right)\).
Answer:
Multiplying this out, we have \(\left(\begin{array}{ll}
1 & 2 \\
k & 1
\end{array}\right)\left(\begin{array}{c}
3 \\
-1
\end{array}\right)\)=\(\left(\begin{array}{c}
\mathbf{1} \\
3 \boldsymbol{k}-\mathbf{1}
\end{array}\right)\)=\(\left(\begin{array}{c}
1 \\
11
\end{array}\right)\), so 3k-1=11, and thus, k=4.

Exercise 3.
Find a matrix \(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\) so that we can represent the transformation L(x,y)=(2x-3y,3x+2y) by \(\).
Answer:
The matrix is \(\left(\begin{array}{cc}
2 & -3 \\
3 & 2
\end{array}\right)\).

Exercise 4.
If a transformation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)=\(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\) has the geometric effect of rotation and dilation, what do you know about the values a,b,c, and d?
Answer:
Since the transformation L(x,y)=(ax-by,bx+ay) has matrix representation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)=\(\left(\begin{array}{cc}
a & -b \\
b & a
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\), we know that a=d and c=-b.

Exercise 5.
Describe the form of a matrix \(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\) so that the transformation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)=\(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\) has the geometric effect of only dilation by a scale factor r.
Answer:
The transformation that scales by factor r has the form L(x,y)=r(x,y)=(rx,ry)=(rx-0y,0x+ry), so the matrix has the form \(\left(\begin{array}{ll}
\boldsymbol{r} & \mathbf{0} \\
\mathbf{0} & \boldsymbol{r}
\end{array}\right)\).

Exercise 6.
Describe the form of a matrix \(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\) so that the transformation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)=\(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\) has the geometric effect of only rotation by θ. Describe the matrix in terms of θ.
Answer:
The matrix has the form \(\left(\begin{array}{cc}
a & -b \\
b & a
\end{array}\right)\), where arg(a+bi)=θ. Thus, a=cos(θ) and b=sin(θ), so the matrix has the form \(\left(\begin{array}{cc}
\cos (\theta) & -\sin (\theta) \\
\sin (\theta) & \cos (\theta)
\end{array}\right)\).

Exercise 7.
Describe the form of a matrix \(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\) so that the transformation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)=\(\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\) has the geometric effect of rotation by θ and dilation with scale factor r. Describe the matrix in terms of θ and r.
Answer:
The matrix has the form \(\left(\begin{array}{cc}
a & -b \\
b & a
\end{array}\right)\), where arg(a+bi)=θ and r=|a+bi|. Thus, a=r cos(θ) and b=r sin(θ), so the matrix has the form \(\left(\begin{array}{cc}
r \cos (\theta) & -r \sin (\theta) \\
r \sin (\theta) & r \cos (\theta)
\end{array}\right)\).

Exercise 8.
Suppose that we have a transformation L\(\) = \(\).
a. Does this transformation have the geometric effect of rotation and dilation?
Answer:
No, the matrix is not in the form \(\), so this transformation is not a rotation and dilation.

b. Transform each of the points A=\(\left(\begin{array}{l}
0 \\
0
\end{array}\right)\), B = \(\left(\begin{array}{l}
1 \\
0
\end{array}\right)\), C = \(\left(\begin{array}{l}
1 \\
1
\end{array}\right)\), and D = \(\left(\begin{array}{l}
0 \\
1
\end{array}\right)\) and plot the images in the plane shown.
Answer:
Eureka Math Precalculus Module 1 Lesson 21 Exercise Answer Key 26

Exercise 9.
Describe the geometric effect of the transformation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)=\(\left(\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\).
Answer:
This transformation does nothing to the point (x,y) in the plane; it is the identity transformation.

Eureka Math Precalculus Module 1 Lesson 21 Problem Set Answer Key

Question 1.
Perform the indicated multiplication.
a. \(\left(\begin{array}{ll}
1 & 2 \\
4 & 8
\end{array}\right)\left(\begin{array}{c}
3 \\
-2
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
-1 \\
-4
\end{array}\right)\)

b. \(\left(\begin{array}{cc}
3 & 5 \\
-2 & -6
\end{array}\right)\left(\begin{array}{l}
2 \\
4
\end{array}\right)\)
Answer:
\(\left(\begin{array}{c}
26 \\
-28
\end{array}\right)\)

c. \(\left(\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right)\left(\begin{array}{l}
6 \\
8
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
14 \\
-2
\end{array}\right)\)

d. \(\left(\begin{array}{ll}
5 & 7 \\
4 & 9
\end{array}\right)\left(\begin{array}{c}
10 \\
100
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
750 \\
940
\end{array}\right)\)

e. \(\left(\begin{array}{ll}
4 & 2 \\
3 & 7
\end{array}\right)\left(\begin{array}{c}
-3 \\
1
\end{array}\right)\)
Answer:
\(\left(\begin{array}{c}
-10 \\
-2
\end{array}\right)\)

f. \(\left(\begin{array}{ll}
6 & 4 \\
9 & 6
\end{array}\right)\left(\begin{array}{c}
2 \\
-3
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
\mathbf{0} \\
0
\end{array}\right)\)

g. \(\left(\begin{array}{cc}
\cos (\theta) & -\sin (\theta) \\
\sin (\theta) & \cos (\theta)
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
x \cos (\theta)-y \sin (\theta) \\
x \sin (\theta)+y \cos (\theta)
\end{array}\right)\)

h. \(\left(\begin{array}{cc}
\pi & 1 \\
1 & -\pi
\end{array}\right)\left(\begin{array}{c}
10 \\
7
\end{array}\right)\)
Answer:
\(\left(\begin{array}{l}
10 \pi+7 \\
10-7 \pi
\end{array}\right)\)

Question 2.
Find a value of k so that Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 60.
Answer:
We have Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 61, so 4k+15=7 and 16+5k=6. Thus, 4k=-8 and 5k=-10, so
k=-2.

Question 3.
Find values of k and m so that Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 61.1.
Answer:
We have Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 62, so 5k+12=7 and -10+4m=-10. Therefore, k=-1 and m=0.

Question 4.
Find values of k and m so that Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 62.1.
Answer:
Since Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 63, we need to find values of k and m so that k+2m=0 and
-2k+5m=-9. Solving this first equation for k gives k=-2m, and substituting this expression for k into the second equation gives –9=-2(-2m)+5m=9m, so we have m=-1. Then, k=-2m gives k=2. Therefore, k=2 and m=-1.

Question 5.
Write the following transformations using matrix multiplication.
a. L(x,y)=(3x-2y,4x-5y)
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 65.1

b. L(x,y)=(6x+10y,-2x+y)
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 66

c. L(x,y)=(25x+10y,8x-64y)
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 67

d. L(x,y)=(πx-y,-2x+3y)
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 68

e. L(x,y)=(10x,100x)
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 69

f. L(x,y)=(2y,7x)
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 70

Question 6.
Identify whether or not the following transformations have the geometric effect of rotation only, dilation only, rotation and dilation only, or none of these.
a. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 71
Answer:
The matrix Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 72 cannot be written in the form Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 73, because 3≠-5, so this is neither a rotation nor a dilation. The transformation L is not one of the specified types of transformations.

b. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 75
Answer:
This transformation has the geometric effect of dilation by a scale factor of 42.

c. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 76
Answer:
The matrix Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 77 has the form Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 78 with a=-4 and b=2. Therefore, this transformation has the geometric effect of rotation and dilation.

d. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 79
The matrix Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 80 cannot be written in the form Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 81, because -1≠-(-1), so this is neither a rotation nor a dilation. The transformation L is not one of the specified types of transformations.

e. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 81.1
Answer:
The matrix Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 82 cannot be written in the form Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 83, because -7≠7, so this is neither a rotation nor a dilation. The transformation L is not one of the specified types of transformations.

f. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 85
Answer:
We see that Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 86, so this transformation has the geometric effect of dilation by \(\sqrt{2}\) and rotation by \(\frac{\pi}{2}\).

Question 7.
Create a matrix representation of a linear transformation that has the specified geometric effect.
a. Dilation by a factor of 4 and no rotation
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 89

b. Rotation by 180° and no dilation
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 90

c. Rotation by –\(\frac{π}{2}\)rad and dilation by a scale factor of 3
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 91

d. Rotation by 30° and dilation by a scale factor of 4
Answer:
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 92

Question 8.
Identify the geometric effect of the following transformations. Justify your answers.
a. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 93
Answer:
Since Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 94, this transformation has the form
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 95 and, thus, represents counterclockwise rotation by \(\frac{3π}{4}\) with no dilation.

b. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 95.1
Answer:
Since cos(\(\frac{π}{2}\))=0 and sin(\(\frac{π}{2}\))=1, this transformation has the form Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 96 and, thus, represents counterclockwise rotation by \(\frac{π}{2}\) and dilation by a scale factor 5.

c. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 97
Answer:
Since cos(π)=-1 and sin(π)=0, this transformation has the form
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 98 and, thus, represents counterclockwise rotation by π and dilation by a scale factor 10.

d. Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 99
Answer:
Since cos(\(\frac{5π}{3}\))=\(\frac{1}{2}\) and sin(\(\frac{5π}{3}\))=-\(\frac{\sqrt{3}}{2}\), this transformation has the form
Engage NY Math Precalculus Module 1 Lesson 21 Problem Set Answer Key 100 and, thus, represents counterclockwise rotation by \(\frac{5π}{3}\) and dilation with scale factor 12.

Eureka Math Precalculus Module 1 Lesson 21 Exit Ticket Answer Key

Question 1.
Evaluate the product \(\left(\begin{array}{cc}
10 & 2 \\
-8 & -5
\end{array}\right)\left(\begin{array}{c}
3 \\
-2
\end{array}\right)\).
Answer:
\(\left(\begin{array}{cc}
10 & 2 \\
-8 & -5
\end{array}\right)\left(\begin{array}{c}
3 \\
-2
\end{array}\right)\) = \(\left(\begin{array}{c}
30-4 \\
-24+10
\end{array}\right)\)
= \(\left(\begin{array}{c}
26 \\
-14
\end{array}\right)\)

Question 2.
Find a matrix representation of the transformation L(x,y)=(3x+4y,x-2y).
Answer:
L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\) = \(\left(\begin{array}{cc}
3 & 4 \\
1 & -2
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\)

Question 3.
Does the transformation L\(\left(\begin{array}{l}
x \\
y
\end{array}\right)\)= \(\left(\begin{array}{cc}
5 & 2 \\
-2 & 5
\end{array}\right)\left(\begin{array}{l}
x \\
y
\end{array}\right)\) represent a rotation and dilation in the plane? Explain how you know.
Answer:
Yes; this transformation can also be represented as L(x,y)=(5x-(-2)y,-2x+5y), which has the geometric effect of counterclockwise rotation by arg(5-2i) and dilation by |5-2i|.

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