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The question asks to find the image of the point \( (-6, -2) \) under a \( 90^\circ \) counterclockwise rotation about the point \( (-4, 1) \).

Here’s how we solve this step by step:

### Step 1: Translate the points
To simplify the rotation, we first translate the center of rotation \( (-4, 1) \) to the origin \( (0, 0) \). We do this by subtracting \( (-4, 1) \) from both points:

- For the point \( (-6, -2) \):
  \[
  (-6, -2) - (-4, 1) = (-6 + 4, -2 - 1) = (-2, -3)
  \]
  
- The new center of rotation becomes \( (0, 0) \).

### Step 2: Apply the \( 90^\circ \) rotation
A \( 90^\circ \) counterclockwise rotation around the origin transforms any point \( (x, y) \) into \( (-y, x) \). So, for the translated point \( (-2, -3) \):

\[
(-2, -3) \xrightarrow{90^\circ} (3, -2)
\]

### Step 3: Translate back
Now, we translate the point \( (3, -2) \) back by adding the center of rotation \( (-4, 1) \):

\[
(3, -2) + (-4, 1) = (3 - 4, -2 + 1) = (-1, -1)
\]

### Final Answer:
The image of the point \( (-6, -2) \) under a \( 90^\circ \) counterclockwise rotation about the point \( (-4, 1) \) is \( (-1, -1) \).

Would you like more details or have any further questions? 

Here are 5 related questions for further practice:
1. What is the image of the point \( (1, 1) \) under a \( 90^\circ \) rotation about \( (0, 0) \)?
2. How would you find the image of \( (5, 3) \) under a \( 180^\circ \) rotation about \( (2, 2) \)?
3. What is the general formula for rotating a point by \( \theta^\circ \) about any arbitrary point \( (h, k) \)?
4. If you rotate \( (2, 5) \) by \( 270^\circ \) clockwise around \( (0, 0) \), what is the resulting point?
5. What happens if you perform a \( 90^\circ \) clockwise rotation instead of counterclockwise on the point \( (-6, -2) \)?

**Tip:** When performing rotations about a point that isn’t the origin, always translate your system to make calculations easier, and then translate back.

What is the image of the point (-6, -2) under a 90° rotation about the point (-4, 1)?

Explore how to rotate a point around another point in coordinate geometry using a 90° counterclockwise rotation. This example includes a step-by-step guide on translating the points to the origin, applying the rotation, and translating back to find the image of the original point.

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