Question

which sequence of transformations maps opqr onto opqr? a reflection across the x - axis followed by a rotation 90° counterclockwise around the origin a rotation 90° counterclockwise around the origin followed by a translation left 6 units a translation left 1 unit and down 1 unit followed by a rotation 180° around the origin

Explanation:

Step1: Analyze reflection across x - axis

A reflection across the x - axis changes the sign of the y - coordinate of each point. For a point $(x,y)$ it becomes $(x, - y)$.

Step2: Analyze 90° counter - clockwise rotation

The rule for a 90° counter - clockwise rotation around the origin is $(x,y)\to(-y,x)$.
Let's assume a general point $(x,y)$ on $OPQR$. After reflection across the x - axis, it is $(x, - y)$. Then after 90° counter - clockwise rotation, it is $(y,x)$.
Let's check the other transformation sequences:

• For a 90° counter - clockwise rotation first, the rule is $(x,y)\to(-y,x)$. Then a translation left 6 units changes $x$ to $x - 6$. This does not match the transformation from $OPQR$ to $O'P'Q'R'$.
• For a translation left 1 unit and down 1 unit first, $(x,y)\to(x - 1,y - 1)$. Then a 180° rotation around the origin $(x,y)\to(-x,-y)$ which does not match the transformation from $OPQR$ to $O'P'Q'R'$.

Answer:

a reflection across the x - axis followed by a rotation 90° counterclockwise around the origin