Question

1. a point q is at (-3,2). it undergoes a rotation of 270 degrees counterclockwise about the origin, a reflection across the line x = - 1, and then a translation of (x,y)→(x + 1,y - 3). what are the coordinates of the final image of q?

. a) (-3,0)
. b) (-5,0)
. c) (4,-6)
. d) (-2,-5)

Explanation:

Step1: Apply 270 - degree counter - clockwise rotation

The rule for a 270 - degree counter - clockwise rotation about the origin is $(x,y)\to(y, - x)$. For the point $Q(-3,2)$, we have $x=-3$ and $y = 2$. So the new point is $(2,3)$.

Step2: Apply reflection across the line $x=-1$

The formula for reflecting a point $(x,y)$ across the line $x = a$ is $(2a - x,y)$. Here $a=-1$ and the point is $(2,3)$. So $2a - x=2\times(-1)-2=-4$. The new point is $(-4,3)$.

Step3: Apply translation

The translation rule is $(x,y)\to(x + 1,y-3)$. For the point $(-4,3)$, we have $x=-4$ and $y = 3$. So $x+1=-4 + 1=-3$ and $y-3=3-3 = 0$. The final point is $(-3,0)$.

Answer:

A. (-3,0)