1. Problem 6a: Find the composition of transformations $T_{(3,-2)} \circ T_{(1,-1)}$. 2. The rule for composition of translations $T_{(a,b)} \circ T_{(c,d)}$ is: $$T_{(a,b)} \circ T_{(c,d)} = T_{(a+c, b+d)}$$ 3. Applying this rule: $$T_{(3,-2)} \circ T_{(1,-1)} = T_{(3+1, -2-1)} = T_{(4,-3)}$$ 4. Problem 6b: Find the composition $T_{(-4,0)} \circ T_{(-2,5)}$. 5. Using the same rule: $$T_{(-4,0)} \circ T_{(-2,5)} = T_{(-4-2, 0+5)} = T_{(-6,5)}$$ 6. Problem 7: Find the image coordinates after translation. 7. For $T_{(3,-1)}(\triangle ABC)$ with $A(5,0), B(-1,2), C(6,-3)$: $$A' = (5+3, 0-1) = (8,-1)$$ $$B' = (-1+3, 2-1) = (2,1)$$ $$C' = (6+3, -3-1) = (9,-4)$$ 8. For $T_{(-4,0)}(\triangle DEF)$ with $D(3,3), E(-2,3), F(0,2)$: $$D' = (3-4, 3+0) = (-1,3)$$ $$E' = (-2-4, 3+0) = (-6,3)$$ $$F' = (0-4, 2+0) = (-4,2)$$ 9. For $T_{(-10,-5)}(\triangle GHI)$ with $G(0,0), H(3,6), J(12,-1)$: $$G' = (0-10, 0-5) = (-10,-5)$$ $$H' = (3-10, 6-5) = (-7,1)$$ $$J' = (12-10, -1-5) = (2,-6)$$ 10. Problem 8a: Find vertices of $r_{180^\circ; O}(\triangle XYZ)$ with $X(-4,7), Y(0,8), Z(2,-1)$. 11. Rotation by $180^\circ$ about origin transforms $(x,y)$ to $(-x,-y)$: $$X' = (4,-7), Y' = (0,-8), Z' = (-2,1)$$ 12. Problem 8b: Find vertices of $r_{270^\circ; O}(\triangle XYZ)$. 13. Rotation by $270^\circ$ about origin transforms $(x,y)$ to $(y,-x)$: $$X' = (7,4), Y' = (8,0), Z' = (-1,-2)$$ 14. Problem 9: Choose the angle of rotational symmetry for the given star-shaped figure. 15. A star shape typically has rotational symmetry of $72^\circ$ or multiples, but given options are 90, 180, 270, or all. 16. Among these, the star has rotational symmetry of $180^\circ$ (option B). Final answers: - 6a: $T_{(4,-3)}$ - 6b: $T_{(-6,5)}$ - 7: $A'(8,-1), B'(2,1), C'(9,-4); D'(-1,3), E'(-6,3), F'(-4,2); G'(-10,-5), H'(-7,1), J'(2,-6)$ - 8a: $X'(4,-7), Y'(0,-8), Z'(-2,1)$ - 8b: $X'(7,4), Y'(8,0), Z'(-1,-2)$ - 9: $180^\circ$ (option B) Regarding photo limits: I do not have a limit on the number of photos you can send.