Question

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 \((x,y)\) on 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 \((x,y)\to(-y,x)\).

Combining these transformations:
First, \((x,y)\to(x + 6,y)\), then \((x + 6,y)\to(x + 6,-y)\), and finally \((x + 6,-y)\to(y,x + 6)\)

Step2: Find an alternative sequence

Let's try a different sequence:
- **Rotate \(90^{\circ}\) counter - clockwise around the origin**: \((x,y)\to(-y,x)\)
- **Translate 6 units right**: \((-y,x)\to(-y+6,x)\)
- **Reflect across the \(x\) - axis**: \((-y + 6,x)\to(-y + 6,-x)\)

Let's check with a sample point. Suppose a point on \(F\) is \((-4,2)\)
- **Original sequence**:
- After translation: \((-4+6,2)=(2,2)\)
- After reflection: \((2,-2)\)
- After rotation: \((2,2)\to(2,2)\to(2,-2)\to(2,2)\) (This is incorrect)
- **New sequence**:
- After rotation: \((-2,-4)\)
- After translation: \((-2 + 6,-4)=(4,-4)\)
- After reflection: \((4,4)\) (Incorrect)

Let's try another sequence:
- **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 take a point \((-4,2)\) on \(F\)
- After reflection: \((-4,-2)\)
- After translation: \((-4+6,-2)=(2,-2)\)
- After rotation: \((2,-2)\to(2,-2)\to(2,-2)\to(2,2)\) (Incorrect)

Let's try:
- **Rotate \(90^{\circ}\) clockwise around the origin**: The rule is \((x,y)\to(y,-x)\)
- **Translate 6 units right**: \((y,-x)\to(y + 6,-x)\)
- **Reflect across the \(x\) - axis**: \((y + 6,-x)\to(y + 6,x)\)

Take the point \((-4,2)\)
- After rotation: \((2,4)\)
- After translation: \((2+6,4)=(8,4)\) (Incorrect)

Let's try:
- **Translate 6 units right**: \((x,y)\to(x + 6,y)\)
- **Rotate \(90^{\circ}\) clockwise around the origin**: \((x + 6,y)\to(y,-(x + 6))\)
- **Reflect across the \(x\) - axis**: \((y,-(x + 6))\to(y,x + 6)\)

Take \((-4,2)\)
- After translation: \((-4 + 6,2)=(2,2)\)
- After rotation: \((2,-2)\)
- After reflection: \((2,2)\) (Incorrect)

Let's try:
- **Reflect across the \(y\) - axis**: \((x,y)\to(-x,y)\)
- **Translate 6 units right**: \((-x,y)\to(-x + 6,y)\)
- **Rotate \(90^{\circ}\) clockwise around the origin**: \((-x + 6,y)\to(y,x - 6)\)

Take \((-4,2)\)
- After reflection: \((4,2)\)
- After translation: \((4+6,2)=(10,2)\) (Incorrect)

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

Take \((-4,2)\)
- After rotation: \((-2,-4)\)
- After reflection: \((-2,4)\)
- After translation: \((4,4)\) (Incorrect)

Let's try:
- **Translate 6 units right**: \((x,y)\to(x + 6,y)\)
- **Rotate \(270^{\circ}\) counter - clockwise (or \(90^{\circ}\) clockwise) around the origin**: \((x + 6,y)\to(y,-(x + 6))\)
- **Reflect across the \(x\) - axis**: \((y,-(x + 6))\to(y,x + 6)\)

Take \((-4,2)\)
- After translation: \((2,2)\)
- After rotation: \((2,-2)\)
- After reflection: \((2,2)\) (Incorrect)

Let's use the property of transformation composition.
The correct sequence is:
- **Reflect across the \(x\) - axis**: \((x,y)\to(x,-y)\)
- **Rotate \(90^{\circ}\) counter - clockwise around the origin**: \((x,-y)\to(y,x)\)
- **Translate 6 units right**: \((y,x)\to(y + 6,x)\)

Take a point \((-4,2)\) on \(F\)
- After reflection: \((-4,-2)\)
- After rotation: \((-2,-4)\)
- After translation: \((4,-4)\)

Answer:

Reflect across the \(x\) - axis, rotate \(90^{\circ}\) counter - clockwise around the origin, then translate 6 units right.