Question

1. follow the sequence of transformations listed.

if the original does not map onto the image,
give a sequence that will take f to f.
translate 6 units right.
reflect across the x - axis.
rotate 90° counterclockwise
around the origin.

Explanation:

Step1: Analyze the given transformations

Let's assume a general point \((x,y)\) in figure \(F\).

• Translation 6 units right: The transformation rule is \((x,y)\to(x + 6,y)\).
• Reflection across the \(x -\)axis: The transformation rule is \((x,y)\to(x,-y)\).
• Rotation \(90^{\circ}\) counter - clockwise around the origin: The transformation rule is \((x,y)\to(-y,x)\).

Step2: Check the composition of transformations

Let's take a vertex of \(F\). Suppose a vertex is \((x,y)\).
First transformation: \((x,y)\to(x + 6,y)\).
Second transformation: \((x+6,y)\to(x + 6,-y)\).
Third transformation: \((x + 6,-y)\to(y,x + 6)\).

Let's try a new sequence.

• Rotation \(90^{\circ}\) counter - clockwise around the origin: \((x,y)\to(-y,x)\).
• Translation 6 units right: \((-y,x)\to(-y+6,x)\).
• Reflection across the \(x -\)axis: \((-y + 6,x)\to(-y + 6,-x)\).

Another approach:

• Reflect across the \(x -\)axis: \((x,y)\to(x,-y)\).
• Translate 6 units right: \((x,-y)\to(x + 6,-y)\).
• Rotate \(90^{\circ}\) counter - clockwise around the origin: \((x + 6,-y)\to(y,x + 6)\).

Let's use the correct sequence:

• Rotate \(90^{\circ}\) counter - clockwise around the origin: For a point \((x,y)\) in \(F\), the transformation is \((x,y)\to(-y,x)\).
• Translate 6 units right: \((-y,x)\to(-y + 6,x)\).
• Reflect across the \(x -\)axis: \((-y+6,x)\to(-y + 6,-x)\).

Answer:

A correct sequence of transformations is: Rotate \(90^{\circ}\) counter - clockwise around the origin, then translate 6 units right, and then reflect across the \(x -\)axis.