Question

what are the coordinates of the vertices of \\( \triangle a ^ { \prime } b ^ { \prime } c ^ { \prime } \\) after a reflection across a line through point p with a y - intercept at \\( y = - 2 \\) followed by translation \\( t _ { \langle 3,3 \
angle } \\) ?
the coordinates of \\( a ^ { \prime } \\) are \\( \square \\).
(type an ordered pair.)

Explanation:

Step1: Determine the coordinates of point A

From the graph, the coordinates of point \(A\) are \((1,7)\).

Step2: Reflect point A across the line \(y = - 2\)

The formula for reflecting a point \((x,y)\) across the line \(y = k\) is \((x,2k - y)\). Here \(k=-2\), so for point \(A(1,7)\), the reflected point \(A_1\) has coordinates \((1,2\times(- 2)-7)=(1,-4 - 7)=(1,-11)\).

Step3: Translate the reflected point \(A_1\) using \(T_{\langle3,3

angle}\)
The translation rule \(T_{\langle a,b
angle}\) for a point \((x,y)\) is \((x + a,y + b)\). For \(A_1(1,-11)\) and \(a = 3,b = 3\), we get \((1+3,-11 + 3)=(4,-8)\).

Answer:

\((4,-8)\)