Question

which sequence of transformations maps ijkl onto ijkl? a translation down 8 units followed by a reflection across the y - axis a rotation 90° counterclockwise around the origin followed by a reflection across the x - axis a translation left 10 units followed by a rotation 90° counterclockwise around the origin

Explanation:

Step1: Identify vertex coordinates

Original vertices: $I(7,4)$, $J(5,2)$, $K(3,5)$, $L(6,6)$
Image vertices: $I'(-4,-1)$, $J'(-2,-3)$, $K'(-5,-1)$, $L'(-6,-4)$

Step2: Test rotation first

90° counterclockwise rotation rule: $(x,y)\to(-y,x)$
Apply to original points:
$I(7,4)\to(-4,7)$, $J(5,2)\to(-2,5)$, $K(3,5)\to(-5,3)$, $L(6,6)\to(-6,6)$

Step3: Test reflection over x-axis

Reflection over x-axis rule: $(x,y)\to(x,-y)$
Apply to rotated points:
$(-4,7)\to(-4,-7)$, $(-2,5)\to(-2,-5)$, $(-5,3)\to(-5,-3)$, $(-6,6)\to(-6,-6)$

Step4: Test translation down 8 units

Translation down 8 units rule: $(x,y)\to(x,y-8)$
Apply to reflected points:
$(-4,-7)\to(-4,-1)$, $(-2,-5)\to(-2,-3)$, $(-5,-3)\to(-5,-1)$, $(-6,-6)\to(-6,-4)$
These match the image vertices.

Step5: Eliminate other options

• Option1: Translation down + y-reflection does not yield the image.
• Option3: Translation left + rotation does not yield the image.

Answer:

a rotation $90^\circ$ counterclockwise around the origin followed by a reflection across the x-axis