Question

mapping figures using transformations
quiz complete
1 2 3 4 5 6 7 8 9
a trapezoid has vertices a(-5, 4), b(-3, 2), c(-7, -2), and d(-7, 2). it is transformed according to the rule $r_y$ ∘ $t_{(-2, 3)}$. what are the coordinates of $c$?
○ (-9, 5)
○ (-9, -1)
○ (5, 1)
○ (9, 1)

Explanation:

Step1: Apply Translation \( T_{(-2, 3)} \) to \( C(-7, -2) \)

The translation rule \( T_{(h, k)} \) is \( (x, y) \to (x + h, y + k) \). Here, \( h = -2 \), \( k = 3 \). So for \( C(-7, -2) \):
\( x' = -7 + (-2) = -9 \)
\( y' = -2 + 3 = 1 \)
After translation, the point is \( (-9, 1) \). Wait, no, wait—the composition is \( r_y \circ T_{(-2, 3)} \), so we first translate, then reflect over \( y \)-axis. Wait, I messed up the order. Composition \( r_y \circ T_{(h,k)} \) means first apply \( T_{(h,k)} \), then apply \( r_y \) (reflection over \( y \)-axis, rule \( (x, y) \to (-x, y) \)).

Wait, let's correct: Step1: Translate \( C(-7, -2) \) by \( T_{(-2, 3)} \): \( (x + (-2), y + 3) = (-7 - 2, -2 + 3) = (-9, 1) \).

Step2: Reflect the translated point over \( y \)-axis: \( r_y \) rule is \( (x, y) \to (-x, y) \). So for \( (-9, 1) \), apply \( r_y \): \( (-(-9), 1) = (9, 1) \)? Wait, no, the options have (9,1) as an option. Wait, maybe the translation is \( T_{(-2, 3)} \) first, then reflection? Wait, the rule is \( r_y \circ T_{(-2, 3)} \), which is "reflect after translating". Wait, let's check the coordinates again.

Wait, original point \( C(-7, -2) \).

First, apply translation \( T_{(-2, 3)} \): \( x = -7 + (-2) = -9 \), \( y = -2 + 3 = 1 \). So translated point is \( (-9, 1) \).

Then apply reflection over \( y \)-axis: \( (x, y) \to (-x, y) \). So \( -(-9) = 9 \), \( y = 1 \). So the result is \( (9, 1) \), which is one of the options (the last option: (9,1)).

Wait, let's verify again:

1. Translation: \( T_{(h, k)} \) is \( (x + h, y + k) \). Here, \( h = -2 \), \( k = 3 \). So \( x = -7 + (-2) = -9 \), \( y = -2 + 3 = 1 \). Translated point: (-9, 1).

1. Reflection over \( y \)-axis: \( (x, y) \to (-x, y) \). So \( -(-9) = 9 \), \( y = 1 \). So \( (9, 1) \), which is option D (if options are as given: last option is (9,1)).

Wait, the options are:

(-9,5), (-9,-1), (5,1), (9,1). So (9,1) is the last option. So that's the answer.

Wait, maybe I misread the translation vector. Let me check the problem again: "transformed according to the rule \( r_y \circ T_{(-2, 3)} \)". So composition: first \( T_{(-2, 3)} \), then \( r_y \). So steps:

• Translate \( C(-7, -2) \) by \( T_{(-2, 3)} \): \( (-7 - 2, -2 + 3) = (-9, 1) \).

• Reflect over \( y \)-axis: \( (-(-9), 1) = (9, 1) \). Yes, that's correct. So the coordinates of \( C'' \) (wait, the problem says \( C'' \)? Wait, the rule is \( r_y \circ T_{(-2, 3)} \), so first translate (to get \( C' \)), then reflect (to get \( C'' \)). So yes, \( C'' = (9, 1) \).

Answer:

(9, 1) (the last option, which is "9,1" as per the options: "○ (9, 1)")