Question

a point at (6, -4) is translated by the vector (-2, 3), then reflected across the x - axis, and finally rotated 270° counterclockwise about the origin. what is the location of the final image?
. a) (-7,4)
. b) (1,4)
. c) (-4,-1)
. d) (1,-4)
o a) (-7,4)
b) (1,4)
o c) (-4,-1)
o d) (1,-4)

Explanation:

Step1: Perform translation

Add the vector $(-2,3)$ to the point $(6, - 4)$. The new $x$-coordinate is $6+( - 2)=4$, and the new $y$-coordinate is $-4 + 3=-1$. So the point after the first - translation is $(4,-1)$.

Step2: Reflect across the x - axis

When reflecting a point $(x,y)$ across the $x$-axis, the $x$-coordinate remains the same, and the $y$-coordinate changes its sign. So for the point $(4,-1)$, after reflection across the $x$-axis, it becomes $(4,1)$.

Step3: Perform the second translation

Add the vector $(-2,3)$ to the point $(4,1)$. The new $x$-coordinate is $4+( - 2)=2$, and the new $y$-coordinate is $1 + 3 = 4$. So the point after the second - translation is $(2,4)$.

Step4: Rotate 270° counter - clockwise about the origin

The rule for a 270° counter - clockwise rotation about the origin for a point $(x,y)$ is $(y,-x)$. For the point $(2,4)$, after rotation, the new $x$-coordinate is $4$, and the new $y$-coordinate is $-2$.

Answer:

None of the given options are correct. The correct final point is $(4,-2)$.