Question

Which of the following sequences of transformation takes points J(9,1) to J'(-3,7)? A) r0eflection across x-axis and translated (x,y) ( x-2, y-2). B) translated (x,y) ( x-2, y+2) and rotated 270 degrees counterclockwise about the origin. C) rotated 90 degrees counterclockwise about the origin and reflected across the x axis. or D) translated (x,y) (x-2, y+2) and rotated 90 degrees counterclockwise about the origin.

Answer 1

The correct option is D.

Explanation:

It is given that sequences of transformation takes points J(9,1) to J'(-3,7).

Option A:

Reflection across x-axis

`(x,y)\rightarrow (x,-y)`

`J(9,1)\rightarrow J_1(9,-1)`

Then translated,

`(x,y)\rightarrow (x-2,y-2)`

`J_1(9,-1)\rightarrow J'(7,-3)\\eq J'(-3,7)`

Therefore option A is incorrect.

Option B:

Translated,

`(x,y)\rightarrow (x-2,y+2)`

`J(9,1)\rightarrow J_1(7,3)`

Then rotated 270 degrees counterclockwise about the origin.

`(x,y)\rightarrow (y,-x)`

`J_1(7,3)\rightarrow J'(3,-7)\\eq J'(-3,7)`

Therefore option B is incorrect.

Option C:

Rotated 90 degrees counterclockwise about the origin

`(x,y)\rightarrow (-y,x)`

`J(9,1)\rightarrow J_1(-1,9)`

Then reflected across the x axis.

`(x,y)\rightarrow (x,-y)`

`J_1(-1,9)\rightarrow J'(-1,-9)\\eq J'(-3,7)`

Therefore option C is incorrect.

Option D:

Translated,

`(x,y)\rightarrow (x-2,y+2)`

`J(9,1)\rightarrow J_1(7,3)`

Then rotated 90 degrees counterclockwise about the origin.

`(x,y)\rightarrow (-y,x)`

`J_1(7,3)\rightarrow J'(-3,7)`

Therefore option D is correct.

Answer 2

Answer: option D) translated (x,y) (x-2, y+2) and rotated 90 degrees counterclockwise about the origin.

Justification:

1) Translation (x,y) → (x - 2, y + 2) ⇒ (9, 1) → (9 - 2, 1 + 2) = (7, 3)

2) Rotation 90° counterclokwise ⇒ (x, y) → (-y, x) ⇒ (7, 3) → (-3, 7)

So, there you have the demonstration that the translation of J (9,1) according to (x,y) → (x - 2, y + 2), followed by a rotation of 90° counterclockwise produces the image J' (-3, 7).