Question

figure p is the result of a transformation on figure o. which transformation would accomplish this?
answer
a rotation 180° clockwise about the origin
a translation 2 units up
a reflection over the x - axis
a reflection over the y - axis

Explanation:

Step1: Analyze Reflection Over x - axis

A reflection over the \(x\) - axis changes the sign of the \(y\) - coordinate of a point \((x,y)\) to \((x, - y)\). Let's consider the key points of Figure \(O\) and Figure \(P\). If we reflect a point from Figure \(O\) (lower part) over the \(x\) - axis, its \(y\) - coordinate will be negated, and it will move to the upper part (where Figure \(P\) is). For example, if a point in Figure \(O\) is \((x,-y)\), after reflection over \(x\) - axis, it becomes \((x,y)\), which matches the position of the corresponding point in Figure \(P\).

Step2: Analyze Other Transformations

• Rotation \(180^{\circ}\) about the origin: A \(180^{\circ}\) rotation about the origin changes a point \((x,y)\) to \((-x,-y)\). This would move the figure to the opposite quadrant, which is not the case here as Figure \(P\) and Figure \(O\) are symmetric about the \(x\) - axis, not related by a \(180^{\circ}\) rotation.
• Translation 2 units up: A translation would move the figure without changing its orientation or symmetry. But Figure \(P\) and Figure \(O\) are mirror images over the \(x\) - axis, not just a translation.
• Reflection over the \(y\) - axis: A reflection over the \(y\) - axis changes the sign of the \(x\) - coordinate of a point \((x,y)\) to \((-x,y)\). This would move the figure horizontally, but Figure \(P\) and Figure \(O\) are symmetric vertically (about \(x\) - axis), not horizontally (about \(y\) - axis).

Answer:

A reflection over the \(x\) - axis