Question

1. δabc has vertices a(-3,2), b(-1,-4), and c(3,1). after two transformations, the vertices of δabc are a(4,0), b(2,-6), and c(-2,-1). which of the following sequence of transformations maps δabc onto δabc?

a. a translation using the rule (x,y) → (x + 1,y - 2), followed by a rotation 180° about the origin using the rule (x,y) → (-x,-y).
b. a reflection in the y-axis using the rule (x,y) → (-x,y), followed by a translation using the rule (x,y) → (x + 1,y - 2).
c. a reflection in the x-axis using the rule (x,y) → (x,-y), followed by a translation using the rule (x,y) → (x + 1,y - 2).
d. a translation using the rule (x,y) → (x + 7,y - 2), followed by a rotation 90° counterclockwise about the origin using the rule (x,y) → (-y,x).

1. on the graph below, figure 1 is the preimage and figure 2 is the image of figure 1 after a sequence of transformations.

complete the statement below that describes the sequence of transformations that maps figure 1 onto figure 2.
figure 1 maps onto figure 2 by a _____________ (blank #1), followed by a _____________ (blank #2).
choices for blank #1:
a. reflection in the x-axis: (x,y) → (x,-y)
b. reflection in the y-axis: (x,y) → (-x,y)
c. reflection in the line y = x: (x,y) → (y,x)
choices for blank #2:
d. translation: (x,y) → (x + 2,y + 3)
e. translation: (x,y) → (x + 2,y - 3)
f. translation: (x,y) → (x - 2,y - 3)

Explanation:

Question 72

Step 1: Test Option A

• Apply translation \( (x,y)\to(x + 1,y - 2) \) to \( A(-3,2) \): \( (-3+1,2 - 2)=(-2,0) \).
• Apply rotation \( 180^\circ \) (rule \( (x,y)\to(-x,-y) \)): \( (2,0)

eq A''(4,0) \). So A is wrong.

Step 2: Test Option B

• Apply reflection over \( y \)-axis (\( (x,y)\to(-x,y) \)) to \( A(-3,2) \): \( (3,2) \).
• Apply translation \( (x,y)\to(x + 1,y - 2) \): \( (3 + 1,2 - 2)=(4,0) \) (matches \( A'' \)).
• Test \( B(-1,-4) \): Reflection: \( (1,-4) \), Translation: \( (1 + 1,-4 - 2)=(2,-6) \) (matches \( B'' \)).
• Test \( C(3,1) \): Reflection: \( (-3,1) \), Translation: \( (-3 + 1,1 - 2)=(-2,-1) \) (matches \( C'' \)). So B works.

Step 3: Test Option C

• Apply reflection over \( x \)-axis (\( (x,y)\to(x,-y) \)) to \( A(-3,2) \): \( (-3,-2) \).
• Apply translation \( (x,y)\to(x + 1,y - 2) \): \( (-3 + 1,-2 - 2)=(-2,-4)

eq A''(4,0) \). So C is wrong.

Step 4: Test Option D

• Apply translation \( (x,y)\to(x + 7,y - 2) \) to \( A(-3,2) \): \( (4,0) \) (matches \( A'' \) for translation).
• Apply rotation \( 90^\circ \) counterclockwise (rule \( (x,y)\to(-y,x) \)): \( (0,4)

eq A''(4,0) \). So D is wrong.

Question 73

Step 1: Analyze Blank #1 (Reflection)

• Observe a point (e.g., top vertex of Figure 1) and its image in Figure 2. A reflection over \( x \)-axis flips \( y \)-coordinate sign, matching the vertical flip. So choose A (reflection over \( x \)-axis).

Step 2: Analyze Blank #2 (Translation)

• After reflection, take a point (e.g., reflected point) and find translation to Figure 2.
• For a point, after reflection, moving \( x \) by \( +2 \) and \( y \) by \( -3 \) matches. So choose E (\( (x,y)\to(x + 2,y - 3) \)).

Answer:

B. A reflection in the \( y \)-axis using the rule \( (x,y) \to (-x,y) \), followed by a translation using the rule \( (x,y) \to (x + 1,y - 2) \).