# 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)).
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. $$\left(\begin{array}{cc} 3 & -2 \\ -1 & 4 \end{array}\right)\left(\begin{array}{l} 1 \\ 5 \end{array}\right)$$
$$\left(\begin{array}{c} 3-10 \\ -1+20 \end{array}\right)$$ = $$\left(\begin{array}{c} -7 \\ 19 \end{array}\right)$$

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

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

Exercise 2.
Find a value of k so that $$\left(\begin{array}{ll} 1 & 2 \\ k & 1 \end{array}\right)\left(\begin{array}{c} 3 \\ -1 \end{array}\right)$$=$$\left(\begin{array}{c} 1 \\ 11 \end{array}\right)$$.
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 .
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?
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.
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.
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.
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?
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. Exercise 9.
Describe the geometric effect of the transformation L$$\left(\begin{array}{l} x \\ y \end{array}\right)$$=$$\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)$$.
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. $$\left(\begin{array}{ll} 1 & 2 \\ 4 & 8 \end{array}\right)\left(\begin{array}{c} 3 \\ -2 \end{array}\right)$$
$$\left(\begin{array}{l} -1 \\ -4 \end{array}\right)$$

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

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

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

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

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

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

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

Question 2.
Find a value of k so that .
We have , 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 .
We have , so 5k+12=7 and -10+4m=-10. Therefore, k=-1 and m=0.

Question 4.
Find values of k and m so that .
Since , 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) b. L(x,y)=(6x+10y,-2x+y) c. L(x,y)=(25x+10y,8x-64y) d. L(x,y)=(πx-y,-2x+3y) e. L(x,y)=(10x,100x) f. L(x,y)=(2y,7x) 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. The matrix cannot be written in the form , 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. This transformation has the geometric effect of dilation by a scale factor of 42.

c. The matrix has the form with a=-4 and b=2. Therefore, this transformation has the geometric effect of rotation and dilation.

d. The matrix cannot be written in the form , 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. The matrix cannot be written in the form , 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. We see that , 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 b. Rotation by 180° and no dilation c. Rotation by –$$\frac{π}{2}$$rad and dilation by a scale factor of 3 d. Rotation by 30° and dilation by a scale factor of 4 Question 8.
a. Since , this transformation has the form and, thus, represents counterclockwise rotation by $$\frac{3π}{4}$$ with no dilation.

b. Since cos($$\frac{π}{2}$$)=0 and sin($$\frac{π}{2}$$)=1, this transformation has the form and, thus, represents counterclockwise rotation by $$\frac{π}{2}$$ and dilation by a scale factor 5.

c. Since cos(π)=-1 and sin(π)=0, this transformation has the form and, thus, represents counterclockwise rotation by π and dilation by a scale factor 10.

d. Since cos($$\frac{5π}{3}$$)=$$\frac{1}{2}$$ and sin($$\frac{5π}{3}$$)=-$$\frac{\sqrt{3}}{2}$$, this transformation has the form 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 $$\left(\begin{array}{cc} 10 & 2 \\ -8 & -5 \end{array}\right)\left(\begin{array}{c} 3 \\ -2 \end{array}\right)$$.
$$\left(\begin{array}{cc} 10 & 2 \\ -8 & -5 \end{array}\right)\left(\begin{array}{c} 3 \\ -2 \end{array}\right)$$ = $$\left(\begin{array}{c} 30-4 \\ -24+10 \end{array}\right)$$
= $$\left(\begin{array}{c} 26 \\ -14 \end{array}\right)$$

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