Question

1. a point at (-5,-2) is rotated 180 counter - clockwise about the origin, then reflected across the line y=-3, and finally translated by the rule (x,y)→(x + 1,y + 5). what is the location of the final image?

. a) (6,3)
. b) (6,-3)
. c) (-4,-7)
. d) (6,2)
○ a) (6,3)
○ b) (6,-3)
○ c) (-4,-7)
○ d) (6,2)

Explanation:

Step1: Apply 180 - degree rotation

When a point $(x,y)$ is rotated 180 - degrees counter - clockwise about the origin, the transformation rule is $(x,y)\to(-x,-y)$. For the point $(-5,-2)$, after rotation, it becomes $(5,2)$.

Step2: Apply reflection across $y = - 3$

The distance between the point $(5,2)$ and the line $y=-3$ is $2-(-3)=5$. After reflection across the line $y = - 3$, the $x$ - coordinate remains the same, and the $y$ - coordinate changes as follows: Let the original $y$ - coordinate be $y_1 = 2$, and the new $y$ - coordinate $y_2=-3-(2 - (-3))=-3 - 5=-8$. So the point after reflection is $(5,-8)$.

Step3: Apply translation

The translation rule is $(x,y)\to(x + 1,y + 5)$. For the point $(5,-8)$, the new $x$ - coordinate is $x_3=5 + 1=6$, and the new $y$ - coordinate is $y_3=-8 + 5=-3$. So the final point is $(6,-3)$.

Answer:

B. $(6,-3)$