Question

A) A quadrilateral with vertices , , , and is rotated about the
origin 270 degrees (counterclockwise). Apply rotation mapping rules to find the image of . (1 point)
G(−10, 1) E(−6, −4) O(2, 0) M(0, 4)
M
M′
Question 12
Question 13
The mapping for a 90-degree counterclockwise rotation is Option # .
A)
A figure is rotated 90 degrees counterclockwise about the origin. Which of the following function
mappings was applied? Enter the number of the correct option.
Option #1:
Option #2:
Option #3:
Option #4:
(1 point)
(x, y) → (y, −x)
(x, y) → (−y, x)
(x, y) → (−x, −y)
(x, y) → (y, x)

Answer 1

To address the questions you've posed, let's analyze them step by step.

### Question A:
You want to find the image of point \( M(0, 4) \) when it is rotated 270 degrees counterclockwise about the origin.

The rules for rotating a point \( (x, y) \) are as follows:
- For a **90-degree counterclockwise rotation**, the new coordinates become \( (-y, x) \).
- For a **180-degree rotation**, the new coordinates become \( (-x, -y) \).
- For a **270-degree counterclockwise rotation**, which is equivalent to a 90-degree clockwise rotation, the new coordinates become \( (y, -x) \).

### Finding \( M' \):
For the point \( M(0, 4) \):
- Using the formula for a 270-degree counterclockwise rotation:
\[
M'(x', y') = (y, -x) = (4, -0) = (4, 0)
\]

Thus, the image of point \( M(0, 4) \) after a 270-degree counterclockwise rotation is \( M'(4, 0) \).

---

### Questions 12 and 13:
In this part, you're looking for the correct mapping function for a 90-degree counterclockwise rotation.

As stated:
- The mapping for a 90-degree counterclockwise rotation is:
\[
(x, y) \rightarrow (-y, x)
\]

Now let’s analyze your options:
- **Option #1:** \( (x, y) \rightarrow (y, -x) \) → Not correct
- **Option #2:** \( (x, y) \rightarrow (-y, x) \) → This is the correct mapping!
- **Option #3:** \( (x, y) \rightarrow (-x, -y) \) → Not correct (180-degree rotation)
- **Option #4:** \( (x, y) \rightarrow (y, x) \) → Not correct

### Final Answers:
- The image of point \( M \) after rotation is \( M'(4, 0) \).
- For the 90-degree counterclockwise rotation, the correct option is **Option #2: \( (x, y) \rightarrow (-y, x) \)**.