Question

figure abcd is reflected across the y-axis, translated 5 units left and 3 units down. the resulting figure is then rotated clockwise about the origin through 90°. find the coordinates of the vertices of the transformed figure.
image of a coordinate grid with figure abcd
option 1: a(-2, -1), b(0, 0), c(-1, -2), and d(1, 1)
option 2: a(-2, 1), b(0, 0), c(1, -2), and d(-1, -1)

Explanation:

Step1: Identify Original Coordinates

From the graph, find original vertices:
- \( A(-4, 1) \), \( B(-5, 3) \)? Wait, no—wait, looking at the grid: Let's recheck. Wait, the x-axis: left is negative. Let's see the points:
- \( A \) is at \( (-4, 1) \)? Wait, no, the grid lines: each square is 1 unit. Let's list correct original coordinates:
- \( A(-4, 1) \), \( B(-5, 3) \)? Wait, no, the red points:
Wait, \( A \) is at \( (-4, 1) \) (x=-4, y=1), \( B \) at \( (-5, 3) \)? Wait, no, maybe I misread. Wait, the options suggest later steps, but let's do transformations step by step.

Step2: Reflect Over y-axis

Reflection over y-axis: \( (x, y) \to (-x, y) \).
- \( A(-4, 1) \to A_1(4, 1) \)
- \( B(-5, 3) \to B_1(5, 3) \)? Wait, no, maybe original coordinates are different. Wait, maybe the original points are:
Wait, looking at the graph: \( A \) is at \( (-4, 1) \), \( B \) at \( (-5, 3) \)? No, maybe \( A(-4, 1) \), \( B(-5, 3) \), \( C(-7, 4) \), \( D(-6, 2) \)? Wait, no, the options have smaller numbers. Wait, maybe I made a mistake. Let's check the options. The options have \( A'(-2, -1) \) etc. Let's reverse-engineer.

Wait, the final transformation is rotation 90° clockwise about origin: \( (x, y) \to (y, -x) \). Let's check the options. Let's take the first option: \( A'(-2, -1) \). If we reverse the rotation (rotate 90° counterclockwise: \( (x, y) \to (-y, x) \)), then before rotation, the point would be \( (-(-1), -2) = (1, -2) \)? No, maybe better to do forward steps.

Wait, let's start over. Let's find original coordinates correctly. From the graph:
- \( A \): x=-4, y=1 (since it's at x=-4, y=1)
- \( B \): x=-5, y=3? No, wait, the blue line: \( A \) is at (-4,1), \( B \) at (-5,3)? No, maybe \( A(-4, 1) \), \( B(-5, 3) \), \( C(-7, 4) \), \( D(-6, 2) \). Wait, no, the options have smaller numbers, so maybe the original coordinates are different. Wait, maybe the original points are \( A(-4, 1) \), \( B(-5, 3) \), \( C(-7, 4) \), \( D(-6, 2) \), but that doesn't match. Wait, maybe the original points are \( A(-4, 1) \), \( B(-5, 3) \), no—wait, the options have \( A'(-2, -1) \), so let's do the transformations step by step with the correct original coordinates.

Wait, maybe the original coordinates are:
- \( A(-4, 1) \)
- \( B(-5, 3) \)
- \( C(-7, 4) \)
- \( D(-6, 2) \)

**Step 1: Reflect over y-axis**
\( (x, y) \to (-x, y) \):
- \( A(-4,1) \to (4,1) \)
- \( B(-5,3) \to (5,3) \)
- \( C(-7,4) \to (7,4) \)
- \( D(-6,2) \to (6,2) \)

**Step 2: Translate 5 units left, 3 units down**
Translation: \( (x, y) \to (x - 5, y - 3) \):
- \( A_1(4,1) \to (4 - 5, 1 - 3) = (-1, -2) \)
- \( B_1(5,3) \to (5 - 5, 3 - 3) = (0, 0) \)
- \( C_1(7,4) \to (7 - 5, 4 - 3) = (2, 1) \)
- \( D_1(6,2) \to (6 - 5, 2 - 3) = (1, -1) \)

**Step 3: Rotate 90° clockwise about origin**
Rotation 90° clockwise: \( (x, y) \to (y, -x) \):
- \( A_2(-1, -2) \to (-2, 1) \)? Wait, no: \( (x, y) \to (y, -x) \). So \( (-1, -2) \to (-2, 1) \)? Wait, no: \( x=-1, y=-2 \), so \( (y, -x) = (-2, 1) \).
- \( B_2(0, 0) \to (0, 0) \) (since \( y=0, -x=0 \))
- \( C_2(2, 1) \to (1, -2) \) ( \( y=1, -x=-2 \))
- \( D_2(1, -1) \to (-1, -1) \) ( \( y=-1, -x=-1 \))

Wait, this matches the second option: \( A'(-2, 1) \), \( B'(0, 0) \), \( C'(1, -2) \), \( D'(-1, -1) \). Wait, but let's check again. Maybe my original coordinates were wrong. Let's re-express:

Alternative original coordinates (maybe I misread the graph):
Suppose original \( A(-4, 1) \), \( B(-5, 3) \), \( C(-7, 4) \), \( D(-6, 2) \). After reflection: \( (4,1), (5,3), (7,4), (6,2) \). Translate 5 left: \( 4-5=-1, 5-5=0, 7-5=2, 6-5=1 \); 3 down: \( 1-3=-2, 3-3=0, 4-3=1, 2-3=-1 \). So after translation: \( A(-1, -2) \), \( B(0, 0) \), \( C(2, 1) \), \( D(1, -1) \). Rotate 90° clockwise: \( (x,y) \to (y, -x) \). So:
- \( A(-1, -2) \to (-2, 1) \) ( \( y=-2, -x=1 \)? Wait, no: \( (x,y) \to (y, -x) \). So \( x=-1, y=-2 \), so \( (y, -x) = (-2, 1) \).
- \( B(0,0) \to (0,0) \)
- \( C(2,1) \to (1, -2) \) ( \( y=1, -x=-2 \))
- \( D(1,-1) \to (-1, -1) \) ( \( y=-1, -x=-1 \))

Yes, this matches the second option. So the correct answer is the second option.

Answer:

A'(-2, 1), B'(0, 0), C'(1, -2), and D'(-1, -1) (the second option, e.g., if options are labeled, but based on the given options, the correct one is the second choice with these coordinates)