Question

give the coordinates after the rotation.
original coordinates → final coordinates
x(-6, 2) → x
y(-4, -2) → y
z(-1, 6) → z
(b) choose the general rule below that describes the rotation mapping △xyz to △xyz.
○ (x, y) → (-y, x) ○ (x, y) → (-y, -x)
○ (x, y) → (x, -y) ○ (x, y) → (y, -x)
○ (x, y) → (-x, y) ○ (x, y) → (y, x)
○ (x, y) → (-x, -y)

Explanation:

Part (a): Finding Coordinates After Rotation

First, we use the general rule identified in part (b), which is \((x, y) \to (y, -x)\).

For \( X(-6, 2) \):

Step 1: Apply the rule \((x, y) \to (y, -x)\)

Substitute \( x = -6 \) and \( y = 2 \).
New \( x \)-coordinate: \( y = 2 \)
New \( y \)-coordinate: \( -x = -(-6) = 6 \)
So, \( X' = (2, 6) \).

For \( Y(-4, -2) \):

Step 1: Apply the rule \((x, y) \to (y, -x)\)

Substitute \( x = -4 \) and \( y = -2 \).
New \( x \)-coordinate: \( y = -2 \)
New \( y \)-coordinate: \( -x = -(-4) = 4 \)
So, \( Y' = (-2, 4) \).

For \( Z(-1, 6) \):

Step 1: Apply the rule \((x, y) \to (y, -x)\)

Substitute \( x = -1 \) and \( y = 6 \).
New \( x \)-coordinate: \( y = 6 \)
New \( y \)-coordinate: \( -x = -(-1) = 1 \)
So, \( Z' = (6, 1) \).

Part (b): Identifying the General Rule

We test each rule with one point (e.g., \( X(-6, 2) \)):

• Rule \((x, y) \to (y, -x)\):

Substitute \( x = -6 \), \( y = 2 \): \( (2, -(-6)) = (2, 6) \), which matches the graph’s \( X' \).

Other rules fail:

• \((x, y) \to (-y, x)\): \( (-2, -6) \) (incorrect).
• \((x, y) \to (-y, -x)\): \( (-2, 6) \) (incorrect).
• \((x, y) \to (x, -y)\): \( (-6, -2) \) (incorrect).
• \((x, y) \to (-x, y)\): \( (6, 2) \) (incorrect).
• \((x, y) \to (y, x)\): \( (2, -6) \) (incorrect).
• \((x, y) \to (-x, -y)\): \( (6, -2) \) (incorrect).

Answer:

s:
(a)
\( X(-6, 2) \to X'(2, 6) \)
\( Y(-4, -2) \to Y'(-2, 4) \)
\( Z(-1, 6) \to Z'(6, 1) \)

(b) The general rule is \(\boldsymbol{(x, y) \to (y, -x)}\).