Question Watch Video Show What is the image point of $(-5,2)$ after the transformation $R_{270^{\circ}} \circ T_{2,-5}$ ?

Step 1 ：The problem is asking for the image point of a given point after a certain transformation. The transformation is a composition of a rotation and a translation. The rotation is 270 degrees counterclockwise, and the translation is 2 units to the right and 5 units down.

Step 2 ：To solve this, we first need to apply the translation to the point, and then apply the rotation to the result.

Step 3 ：The translation moves every point in the plane 2 units to the right and 5 units down. So, the image of any point \((x,y)\) under this translation is \((x+2, y-5)\).

Step 4 ：The rotation rotates every point in the plane 270 degrees counterclockwise about the origin. The image of any point \((x,y)\) under this rotation is \((-y, x)\).

Step 5 ：So, to find the image of the point \((-5,2)\) under the transformation \(R_{270^{\circ}} \circ T_{2,-5}\), we first apply the translation to get \((-5+2, 2-5) = (-3,-3)\), and then apply the rotation to get \((-(-3), -3) = (3,-3)\).

Step 6 ：Final Answer: The image point of \((-5,2)\) after the transformation \(R_{270^{\circ}} \circ T_{2,-5}\) is \(\boxed{(3,-3)}\).