Question

identifying transformations
what are the vertices for the final image after applying
the composition ( t _ { - 2,4 } circ r _ { o, 180 ^ { circ } } ) to ( delta x y z ) ?
( x ^ { prime prime } ) is
( y ^ { prime prime } ) is
( z ^ { prime prime } ) is

Explanation:

Step1: Find coordinates of \( \triangle XYZ \)

From the graph, \( X(2,5) \), \( Y(2,3) \), \( Z(4,3) \).

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

The rule for \( R_{O,180^{\circ}} \) is \( (x,y)\to(-x,-y) \).
For \( X(2,5) \): \( X'( - 2,-5) \).
For \( Y(2,3) \): \( Y'( - 2,-3) \).
For \( Z(4,3) \): \( Z'( - 4,-3) \).

Step3: Apply \( T_{-2,4} \)

The rule for \( T_{-2,4} \) is \( (x,y)\to(x - 2,y + 4) \).
For \( X'(-2,-5) \): \( X''(-2-2,-5 + 4)=(-4,-1) \).
For \( Y'(-2,-3) \): \( Y''(-2-2,-3 + 4)=(-4,1) \).
For \( Z'(-4,-3) \): \( Z''(-4-2,-3 + 4)=(-6,1) \).

Answer:

\( X'' \) is \( (-4,-1) \), \( Y'' \) is \( (-4,1) \), \( Z'' \) is \( (-6,1) \).