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 unitsa translation left 5 units and then a reflection over the x - axisa 180° rotation using the origin as the center of rotationa 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 (which is a vertical translation) and then reflected over the \(y -\)axis. Let's check each option:

• Option 1: A translation left 5 units and then a translation down 5 units. This is a combination of two translations. A translation left 5 units changes the \(x -\)coordinate (\(x\to x - 5\)) and then a translation down 5 units changes the \(y -\)coordinate (\(y\to y-5\)). This is not the same as the given transformation (translation down 5 and then reflection over \(y -\)axis).
• Option 2: A translation left 5 units and then a reflection over the \(x -\)axis. A translation left 5 units (\(x\to x - 5\)) and then a reflection over the \(x -\)axis (\(y\to - y\)). This is not the same as the given transformation (translation down 5 and then reflection over \(y -\)axis).
• Option 3: A \(180^{\circ}\) rotation using the origin as the center of rotation. A \(180^{\circ}\) rotation about the origin has the transformation rule \((x,y)\to(-x,-y)\). This is not the same as the given transformation (translation down 5 and then reflection over \(y -\)axis).
• 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, a reflection over the \(y -\)axis has the transformation rule \((x,y)\to(-x,y)\).
• Then, a translation down 5 units has the transformation rule \((-x,y)\to(-x,y - 5)\).
• For the given transformation:
• Let's assume a point \((x,y)\) on figure \(X\). First, translation down 5 units: \((x,y)\to(x,y - 5)\). Then reflection over the \(y -\)axis: \((x,y - 5)\to(-x,y - 5)\).

Answer:

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