Question

1. which transformation maps (2, 3) to (-2, -3) after reflecting over both axes?

reflection over x - axis
rotation of 360 degrees
rotation of 180 degrees
translation of 4 left and 6 down

Explanation:

Step1: Analyze reflection over x - axis

When reflecting a point \((x,y)\) over the \(x\) - axis, the transformation rule is \((x,y)\to(x, - y)\). For the point \((2,3)\), reflection over the \(x\) - axis gives \((2,-3)\). Then reflecting \((2,-3)\) over the \(y\) - axis (transformation rule \((x,y)\to(-x,y)\)) gives \((- 2,-3)\)

Step2: Analyze rotation of \(360\) degrees

A rotation of \(360\) degrees about the origin is equivalent to the identity transformation. For a point \((x,y)\), \((x,y)\to(x,y)\) after a \(360\) - degree rotation. Reflecting \((2,3)\) over the \(x\) - axis gives \((2,-3)\), and then reflecting \((2,-3)\) over the \(y\) - axis gives \((-2,-3)\). But rotation of \(360\) degrees does not change the position before reflections, so it is not the correct transformation.

Step3: Analyze rotation of \(180\) degrees

The rule for a \(180\) - degree rotation about the origin is \((x,y)\to(-x,-y)\). Reflecting \((2,3)\) over the \(x\) - axis gives \((2,-3)\), and then reflecting \((2,-3)\) over the \(y\) - axis gives \((-2,-3)\). A \(180\) - degree rotation of \((2,3)\) is \((-2,-3)\) without considering the double - reflection, but we need to check the composition with reflections.

Step4: Analyze translation

The rule for a translation of \(4\) units left and \(6\) units down is \((x,y)\to(x - 4,y-6)\). Reflecting \((2,3)\) over the \(x\) - axis gives \((2,-3)\), then reflecting \((2,-3)\) over the \(y\) - axis gives \((-2,-3)\). A translation of \((2,3)\) by \(4\) left and \(6\) down is \((2-4,3 - 6)=(-2,-3)\) only if we consider the double - reflection first. But the key is the transformation property.

Since reflection over the \(x\) - axis (and then \(y\) - axis) (or vice - versa) for the point \((2,3)\) gives \((-2,-3)\)

Answer:

Reflection over x - axis