Question

if \\(p = (2, -1)\\), find the image of \\(p\\) under the following rotation.

\\(270^\circ\\) counterclockwise about the origin

\\((?, \quad)\\)

enter the number that belongs in the green box.

Explanation:

Identify the given point and rotation

The given point is \(P = (2, -1)\).
We need to rotate this point \(270^\circ\) counterclockwise about the origin.
The target is the \(x\)-coordinate of the image point, which is in the green box containing the question mark.

Apply the rotation rule

A counterclockwise rotation of \(270^\circ\) about the origin maps a point \((x, y)\) to \((y, -x)\).
Let the image point be \(P' = (x', y')\).
Using the rule:

$$ x' = y $$

$$ y' = -x $$

Calculate the coordinates of the image

Substitute \(x = 2\) and \(y = -1\) into the rotation formulas:

$$ x' = -1 $$

$$ y' = -(2) = -2 $$

Thus, the image point is \(P' = (-1, -2)\).

Determine the value in the green box

The green box corresponds to the first coordinate (the \(x\)-coordinate) of the image point \(([?], [ ])\).
Therefore, the value in the green box is \(-1\).

Answer:

Enter the number that belongs in the green box: <blank>-1</blank>