Questions: The graph shows kites WXYZ and W'X'Y'Z'. Which sequence of transformations maps WXYZ onto W'X'Y'Z'? a rotation 90° counterclockwise around the origin followed by a translation up 6 units a rotation 90° clockwise around the origin followed by a reflection across the y-axis a rotation 90° clockwise around the origin followed by a translation right 8 units

The graph shows kites WXYZ and W'X'Y'Z'.

Which sequence of transformations maps WXYZ onto W'X'Y'Z'?
a rotation 90° counterclockwise around the origin followed by a translation up 6 units
a rotation 90° clockwise around the origin followed by a reflection across the y-axis
a rotation 90° clockwise around the origin followed by a translation right 8 units

Transcript text: The graph shows kites $W X Y Z$ and $W^{\prime} X^{\prime} Y^{\prime} Z^{\prime}$. Which sequence of transformations maps $W X Y Z$ onto $W^{\prime} X^{\prime} Y^{\prime} Z^{\prime}$ ? a rotation $90^{\circ}$ counterclockwise around the origin followed by a translation up 6 units a rotation $90^{\circ}$ clockwise around the origin followed by a reflection across the $y$-axis a rotation $90^{\circ}$ clockwise around the origin followed by a translation right 8 units

Solution

Solution Steps

Step 1: Rotate WXYZ 90° clockwise about the origin

When rotating WXYZ 90° clockwise about the origin, the coordinates of the resulting image are: W'(-1, -5) -> W''(5, -1) X'(-3, -4) -> X''(4,-3) Y'(-5,-8) -> Y''(8, -5) Z'(-7,-4) -> Z''(4, -7)

Step 2: Translate the rotated image 8 units to the right

When translating W''X''Y''Z'' 8 units to the right, we get: W''(5, -1) -> W'''(13, -1) X''(4,-3) -> X'''(12, -3) Y''(8, -5) -> Y'''(16, -5) Z''(4, -7) -> Z'''(12, -7)

However, observing the graph, the coordinates of W'X'Y'Z' are W'(6,5), X'(4,3), Y'(2,5) and Z'(4,7).

Step 3: Compare intermediate image with given W'X'Y'Z'

The vertices of the resulting image of the combined transformations in the third option do not match with the given image W'X'Y'Z'. Let's examine the first option.

Rotate WXYZ 90° counter-clockwise and then translate up 6 units:

Rotation: W(-5, -1) -> W''(1, -5) X(-4, -3) -> X''(3, -4) Y(-8, -5) -> Y''(5, -8) Z(-4, -7) -> Z''(7, -4)

Translation: W''(1, -5) -> W'''(1, 1) X''(3, -4) -> X'''(3, 2) Y''(5, -8) -> Y'''(5, -2) Z''(7, -4) -> Z'''(7, 2)

This also doesn't match. What about the second option?

Rotating clockwise 90° matches the first step of option 3, so let's reflect this across the y-axis instead of translating right.

Reflection: W''(5, -1) -> W'''(-5, -1) X''(4, -3) -> X'''(-4, -3) Y''(8, -5) -> Y'''(-8, -5) Z''(4, -7) -> Z'''(-4, -7)

This also does not match W'X'Y'Z'.

Consider what happens if we reflect WXYZ across the y-axis and translate the result up by 6 units. W(-5, -1) -> W'(5,-1) X(-4, -3) -> X'(4,-3) Y(-8, -5) -> Y'(8,-5) Z(-4, -7) -> Z'(4,-7)

W'(5,-1)->W''(5,5) X'(4,-3)->X''(4,3) Y'(8,-5)->Y''(8,1) Z'(4,-7)->Z''(4,-1)

Consider rotating 90 degrees counter-clockwise about the origin and translate 6 units to the right.