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.