Question

part a: the archeologist turns the fragment over, reflecting it over the y - axis. what are the new coordinates of the vertices? the pre - image coordinates of the vertices is (-4,4),(-2,4),(-4,2),(-2,2). the new image coordinates of the vertices is (3,2) 1 part b: the archeologist then rotates the fragment 180° about the origin to see it from a different angle. what are the new coordinates of the vertices?

Explanation:

Step1: Recall reflection over y - axis rule

When reflecting a point $(x,y)$ over the y - axis, the new point is $(-x,y)$.
Let the original vertices be $A(-4,4)$, $B(-2,4)$, $C(-4,2)$, $D(-2,2)$.
For point $A(-4,4)$:
New x - coordinate is $-(-4)=4$, y - coordinate remains 4. So new point is $(4,4)$.
For point $B(-2,4)$:
New x - coordinate is $-(-2)=2$, y - coordinate remains 4. So new point is $(2,4)$.
For point $C(-4,2)$:
New x - coordinate is $-(-4)=4$, y - coordinate remains 2. So new point is $(4,2)$.
For point $D(-2,2)$:
New x - coordinate is $-(-2)=2$, y - coordinate remains 2. So new point is $(2,2)$.

Step2: Recall 180 - degree rotation about origin rule

When rotating a point $(x,y)$ 180 degrees about the origin, the new point is $(-x,-y)$.
For the points after reflection $(4,4)$, $(2,4)$, $(4,2)$, $(2,2)$:
For point $(4,4)$:
New x - coordinate is $-4$, new y - coordinate is $-4$. So new point is $(-4,-4)$.
For point $(2,4)$:
New x - coordinate is $-2$, new y - coordinate is $-4$. So new point is $(-2,-4)$.
For point $(4,2)$:
New x - coordinate is $-4$, new y - coordinate is $-2$. So new point is $(-4,-2)$.
For point $(2,2)$:
New x - coordinate is $-2$, new y - coordinate is $-2$. So new point is $(-2,-2)$.

Answer:

For Part A: $(4,4),(2,4),(4,2),(2,2)$
For Part B: $(-4,-4),(-2,-4),(-4,-2),(-2,-2)$