What is the image point of left parenthesis, minus, 6, comma, minus, 8, right parenthesis(−6,−8) after the transformation R, start subscript, 90, degrees, end subscript, circle, r, start subscript, y, equals, minus, x, end subscriptR 
90 
∘
 
​
 ∘r 
y=−x
​
 ?

To find the image point of \((-6, -8)\) after the transformation \( R_{90^\circ} \circ r_{y=-x} \), we need to break this down step by step.

### Step 1: Reflection across the line \( y = -x \)
The reflection of a point \((x, y)\) across the line \( y = -x \) results in the point \((-y, -x)\).

For the point \((-6, -8)\):
- Reflecting \((-6, -8)\) across \( y = -x \) gives:
  \[
  (-(-8), -(-6)) = (8, 6)
  \]

### Step 2: Rotation by \(90^\circ\) counterclockwise
The point \( (x, y) \) rotated \(90^\circ\) counterclockwise about the origin becomes \((-y, x)\).

For the point \((8, 6)\):
- Rotating \((8, 6)\) \(90^\circ\) counterclockwise gives:
  \[
  (-6, 8)
  \]

### Final Answer:
The image point of \((-6, -8)\) after the transformation \( R_{90^\circ} \circ r_{y=-x} \) is \((-6, 8)\).

Would you like details on any part of this process, or do you have any questions?

Here are 5 related questions you might find interesting:
1. What is the result of applying only the reflection \( r_{y=-x} \) to a general point \((x, y)\)?
2. How does the result differ if we rotate \( (x, y) \) by \( 90^\circ \) clockwise instead of counterclockwise?
3. What is the effect of applying the rotation before the reflection?
4. How does the transformation \( R_{180^\circ} \circ r_{y=-x} \) affect a general point \((x, y)\)?
5. Can we represent the transformation \( R_{90^\circ} \circ r_{y=-x} \) using matrix multiplication?

**Tip:** Visualizing transformations using sketches or coordinate grids can help you understand the steps more clearly.

What is the image point of left parenthesis, minus, 6, comma, minus, 8, right parenthesis(−6,−8) after the transformation R, start subscript, 90, degrees, end subscript, circle, r, start subscript, y, equals, minus, x, end subscriptR 90 ∘ ∘r y=−x?

Learn how to determine the image point of (-6,-8) after applying a 90° counterclockwise rotation followed by a reflection across the line y = -x. This solution covers key concepts in coordinate geometry, suitable for students in Grades 9-11.

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