What is the image point of left parenthesis, minus, 1, comma, 9, right parenthesis(−1,9) after the transformation R, start subscript, 270, degrees, end subscript, circle, r, start subscript, y-axis, end subscriptR 
270 
∘
 
​
 ∘r 
y-axis

The transformation you're asking about is a combination of two operations:

1. **\( R_{270^\circ} \)**: A rotation of 270 degrees counterclockwise around the origin.
2. **\( r_{y\text{-axis}} \)**: A reflection across the y-axis.

Let's break this down step by step:

### Step 1: Rotation by 270 Degrees (\( R_{270^\circ} \))
The general rule for rotating a point \( (x, y) \) counterclockwise by 270 degrees is:
\[
(x', y') = (y, -x)
\]
So, for the point \( (-1, 9) \), after applying a 270° rotation:
\[
(x', y') = (9, 1)
\]

### Step 2: Reflection over the y-axis (\( r_{y\text{-axis}} \))
The rule for reflecting a point \( (x, y) \) over the y-axis is:
\[
(x'', y'') = (-x, y)
\]
After reflecting \( (9, 1) \) over the y-axis, we get:
\[
(x'', y'') = (-9, 1)
\]

### Final Answer:
The image point of \( (-1, 9) \) after the transformation \( R_{270^\circ} \circ r_{y\text{-axis}} \) is \( (-9, 1) \).

Would you like a more detailed explanation of each step?

Here are 5 related questions:
1. What is the rule for rotating a point 90 degrees counterclockwise?
2. How would you reflect a point across the x-axis instead of the y-axis?
3. What is the transformation matrix for a 270-degree rotation?
4. How does reflecting across the origin differ from reflecting across the y-axis?
5. What would happen if we rotated the point by 180 degrees instead?

**Tip**: Rotations by 90°, 180°, and 270° follow specific rules that can be easily memorized using coordinates.

What is the image point of (-1, 9) after the transformation R_{270°} ∘ r_{y-axis}?

Discover how to find the image point of (-1, 9) after a 270° counterclockwise rotation followed by a reflection over the y-axis. This step-by-step solution includes key geometry concepts like rotation and reflection in coordinate transformations. Suitable for Grades 9-11.

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