Question

graph \\( \overline { x y } \\) with endpoints \\( x ( - 3,1 ) \\) and \\( y ( 4, - 5 ) \\) and its image after the composition.
translation: \\( ( x, y ) \
ightarrow ( x, y + 2 ) \\)
rotation: \\( 90 ^ { \circ } \\) counterclockwise about the origin
write a coordinate rule for the composition.
\\( ( x, y ) \
ightarrow ( \square, \square ) \\)

Explanation:

Step1: Apply the translation

For a translation \((x,y)\to(x,y + 2)\), if we have a point \((x,y)\), after translation, the \(y\) - coordinate is increased by \(2\).

Step2: Apply the rotation

The rule for a \(90^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-y,x)\). Let \((x',y')=(x,y + 2)\) be the point after translation. Then after rotation, the new point \((x'',y'')\) is given by substituting \(x'=x\) and \(y'=y + 2\) into the rotation rule.

We get \((x,y)\to(x,y + 2)\to(-(y + 2),x)\)

Answer:

\((x,y)\to(-y - 2,x)\)