Question

question
what is the image point of (0, -6) after the transformation $r_{180^{circ}} \circ t_{-3, -1}$?
answer attempt 1 out of 2

Explanation:

Step1: Apply Translation \( T_{-3,-1} \)

The translation \( T_{a,b} \) transforms a point \( (x,y) \) to \( (x + a, y + b) \). Here, \( a=-3 \), \( b = -1 \), and the original point is \( (0,-6) \). So, after translation:
\( x = 0 + (-3)=-3 \), \( y=-6+(-1)=-7 \). The point becomes \( (-3,-7) \).

Step2: Apply Rotation \( R_{180^\circ} \)

The rule for \( 180^\circ \) rotation about the origin is \( (x,y)\to(-x,-y) \). For the point \( (-3,-7) \):
\( x=-(-3)=3 \), \( y=-(-7)=7 \).

Answer:

\( (3,7) \)