To determine which sequence of transformations produces the same image point [tex]\((2, -1)\)[/tex] as the translation of [tex]\((5,3)\)[/tex] left 3 units and down 4 units, we should examine each sequence of transformations step by step.
Sequence 1: Translate left 2 units and down 3 units, and then reflect over the [tex]\(x\)[/tex]-axis
1. Translate left 2 units and down 3 units:
- Start with point [tex]\((5, 3)\)[/tex].
- Translate: [tex]\( (5 - 2, 3 - 3) = (3, 0) \)[/tex].
2. Reflect over the [tex]\(x\)[/tex]-axis:
- Reflect the point [tex]\((3, 0)\)[/tex].
- Reflection: [tex]\((3, -0) = (3, 0) \)[/tex].
Resulting point: [tex]\((3, 0)\)[/tex].
Sequence 2: Translate left 3 units and down 2 units, and then reflect over the [tex]\(x\)[/tex]-axis
1. Translate left 3 units and down 2 units:
- Start with point [tex]\((5, 3)\)[/tex].
- Translate: [tex]\( (5 - 3, 3 - 2) = (2, 1) \)[/tex].
2. Reflect over the [tex]\(x\)[/tex]-axis:
- Reflect the point [tex]\((2, 1)\)[/tex].
- Reflection: [tex]\((2, -1) \)[/tex].
Resulting point: [tex]\((2, -1)\)[/tex].
Sequence 3: Reflect over the [tex]\(x\)[/tex]-axis, and then translate left 2 units and down 3 units
1. Reflect over the [tex]\(x\)[/tex]-axis:
- Start with point [tex]\((5, 3)\)[/tex].
- Reflection: [tex]\((5, -3) \)[/tex].
2. Translate left 2 units and down 3 units:
- From point [tex]\((5, -3)\)[/tex], translate: [tex]\((5 - 2, -3 - 3) = (3, -6) \)[/tex].
Resulting point: [tex]\((3, -6)\)[/tex].
Sequence 4: Reflect over the [tex]\(x\)[/tex]-axis, and then translate left 3 units and down 2 units
1. Reflect over the [tex]\(x\)[/tex]-axis:
- Start with point [tex]\((5, 3)\)[/tex].
- Reflection: [tex]\((5, -3) \)[/tex].
2. Translate left 3 units and down 2 units:
- From point [tex]\((5, -3)\)[/tex], translate: [tex]\((5 - 3, -3 - 2) = (2, -5) \)[/tex].
Resulting point: [tex]\((2, -5)\)[/tex].
Now, we compare these transformation results with the required image point [tex]\((2, -1)\)[/tex]:
- Sequence 1 results in [tex]\((3, 0)\)[/tex].
- Sequence 2 results in [tex]\((2, -1)\)[/tex].
- Sequence 3 results in [tex]\((3, -6)\)[/tex].
- Sequence 4 results in [tex]\((2, -5)\)[/tex].
The correct sequence that produces the same image point [tex]\((2, -1)\)[/tex] is:
Translate left 3 units and down 2 units, and then reflect over the [tex]\(x\)[/tex]-axis.
