Question

question 20 of 40
which sequence of transformations carries abcd onto efgh?
a. reflection across the x-axis followed by rotation of 90° counterclockwise about the origin
b. translation of three units up followed by reflection across the y-axis
c. rotation of 90° clockwise about the origin followed by reflection across the x-axis
d. reflection across the x-axis followed by reflection across the y-axis

Explanation:

Step1: Identify coordinates of ABCD

Let's list coordinates from the grid:
$A=(2,-1)$, $B=(2,-3)$, $C=(6,-3)$, $D=(6,-1)$

Step2: Test Option A (x-reflect + 90° CCW rotation)

First, reflect across x-axis: $(x,y)\to(x,-y)$
Reflected points: $A'(2,1)$, $B'(2,3)$, $C'(6,3)$, $D'(6,1)$
Then 90° counterclockwise rotation about origin: $(x,y)\to(-y,x)$
Transformed points: $A''(-1,2)$, $B''(-3,2)$, $C''(-3,6)$, $D''(-1,6)$
These do not match EFGH ($E=(-2,1)$, $F=(-2,3)$, $G=(-6,3)$, $H=(-6,1)$)

Step3: Test Option B (up 3 + y-reflect)

First, translate up 3 units: $(x,y)\to(x,y+3)$
Translated points: $A'(2,2)$, $B'(2,0)$, $C'(6,0)$, $D'(6,2)$
Then reflect across y-axis: $(x,y)\to(-x,y)$
Transformed points: $A''(-2,2)$, $B''(-2,0)$, $C''(-6,0)$, $D''(-6,2)$
These do not match EFGH

Step4: Test Option C (90° CW rotation + x-reflect)

First, 90° clockwise rotation about origin: $(x,y)\to(y,-x)$
Rotated points: $A'(-1,-2)$, $B'(-3,-2)$, $C'(-3,-6)$, $D'(-1,-6)$
Then reflect across x-axis: $(x,y)\to(x,-y)$
Transformed points: $A''(-1,2)$, $B''(-3,2)$, $C''(-3,6)$, $D''(-1,6)$
These do not match EFGH

Step5: Test Option D (x-reflect + y-reflect)

First, reflect across x-axis: $(x,y)\to(x,-y)$
Reflected points: $A'(2,1)$, $B'(2,3)$, $C'(6,3)$, $D'(6,1)$
Then reflect across y-axis: $(x,y)\to(-x,y)$
Transformed points: $A''(-2,1)$, $B''(-2,3)$, $C''(-6,3)$, $D''(-6,1)$
These match EFGH ($E=(-2,1)$, $F=(-2,3)$, $G=(-6,3)$, $H=(-6,1)$)

Answer:

D. Reflection across the x-axis followed by reflection across the y-axis