Question

the graph shows rhombuses jklm and jklm. which sequence of transformations maps jklm onto jklm? a rotation 180° around the origin followed by a translation left 8 units and up 10 units a translation right 11 units and up 4 units followed by a reflection across the y - axis a reflection across the y - axis followed by a rotation 90° counterclockwise around the origin

Explanation:

Step1: Analyze rotation rule

A 180 - degree rotation of a point $(x,y)$ around the origin gives $(-x,-y)$. Then a translation left 8 units (subtract 8 from $x$ - coordinate) and up 10 units (add 10 to $y$ - coordinate) will change the point further. But this doesn't match the transformation from $JKLM$ to $J'K'L'M'$.

Step2: Analyze translation - reflection

A translation right 11 units (add 11 to $x$ - coordinate) and up 4 units (add 4 to $y$ - coordinate) of a point $(x,y)$ gives $(x + 11,y + 4)$. Then a reflection across the $y$ - axis (change the sign of the $x$ - coordinate) gives $(-(x + 11),y + 4)$. This doesn't match the transformation from $JKLM$ to $J'K'L'M'$.

Step3: Analyze reflection - rotation

Let's assume a point $(x,y)$ in $JKLM$. A reflection across the $y$ - axis gives $(-x,y)$. Then a 90 - degree counter - clockwise rotation of a point $(x,y)$ around the origin has the rule $(x,y)\to(-y,x)$. For the point $(-x,y)$ after 90 - degree counter - clockwise rotation, we get $(-y,-x)$. By observing the graph, this sequence of transformations maps $JKLM$ onto $J'K'L'M'$.

Answer:

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