Question

figure x is translated down 5 and then reflected over the y - axis, forming figure y.
which sequence of transformations results in the same transformation?
a translation left 5 units and then a translation down 5 units
a translation left 5 units and then a reflection over the x - axis
a 180° rotation using the origin as the center of rotation
a reflection over the y - axis and then a translation down 5 units

Explanation:

Step1: Analyze the given transformation

The original figure \(X\) is translated down \(5\) units and then reflected over the \(y -\)axis to get figure \(Y\). Let's check each option:

• Option 1: A translation left \(5\) units and then a translation down \(5\) units. This is just a combination of two translations. The direction of movement (left - right is \(x\) - direction, up - down is \(y\) - direction) is different from the given transformation (which involves a reflection).
• Option 2: A translation left \(5\) units and then a reflection over the \(x\) - axis. A reflection over the \(x\) - axis changes the \(y\) - coordinate sign (\((x,y)\to(x, - y)\)), while the given transformation has a reflection over the \(y\) - axis (\((x,y)\to(-x,y)\)) after a vertical translation.
• Option 3: A \(180^{\circ}\) rotation using the origin as the center of rotation. A \(180^{\circ}\) rotation about the origin transforms a point \((x,y)\) to \((-x, - y)\). This is different from the given transformation (translation down \(5\) units: \((x,y)\to(x,y - 5)\) and then reflection over \(y\) - axis: \((x,y-5)\to(-x,y - 5)\))
• Option 4: A reflection over the \(y\) - axis and then a translation down \(5\) units. Let \((x,y)\) be a point on figure \(X\). First, reflection over the \(y\) - axis gives \((-x,y)\). Then translation down \(5\) units gives \((-x,y - 5)\), which is the same as the transformation: translation down \(5\) units (\((x,y)\to(x,y - 5)\)) and then reflection over the \(y\) - axis (\((x,y - 5)\to(-x,y - 5)\)) due to the commutative property of these two transformations (for a function \(y = f(x)\), \(y=f(x)-k\) and \(y = f(-x)-k\) can be achieved by first reflecting \(y = f(x)\) over \(y\) - axis (\(y = f(-x)\)) and then translating down \(k\) units or vice - versa)

Answer:

A reflection over the \(y\) - axis and then a translation down \(5\) units.