Answer :
Let's analyze Santana's sequence of transformations step-by-step to determine if his thinking is correct.
### Step 1: Reflection across the y-axis
1. Pre-image Coordinates: (4, 0)
- Santana starts with the point [tex]\((4, 0)\)[/tex], which is the pre-image.
2. Reflection across the y-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the y-axis changes its coordinates to [tex]\((-x, y)\)[/tex].
- Applying this transformation to [tex]\((4, 0)\)[/tex]:
[tex]\[ (4, 0) \xrightarrow{\text{reflect across y-axis}} (-4, 0) \][/tex]
- This gives us the reflected coordinates [tex]\((-4, 0)\)[/tex].
### Step 2: Translation 2 units to the right
3. Translation 2 units to the right:
- Translating a point [tex]\((x, y)\)[/tex] 2 units to the right changes its coordinates to [tex]\((x + 2, y)\)[/tex].
- Applying this transformation to [tex]\((-4, 0)\)[/tex]:
[tex]\[ (-4, 0) \xrightarrow{\text{translate 2 units right}} (-2, 0) \][/tex]
- This gives us the final image coordinates [tex]\((-2, 0)\)[/tex].
### Verification of Santana’s Thinking
4. Initial Pre-image:
- The original pre-image coordinates are [tex]\((4, 0)\)[/tex].
5. Reflection Result:
- Reflecting [tex]\((4, 0)\)[/tex] across the y-axis yields [tex]\((-4, 0)\)[/tex].
6. Translation Result:
- Translating [tex]\((-4, 0)\)[/tex] 2 units to the right yields [tex]\((-2, 0)\)[/tex].
### Final Answer
The sequence of transformations suggested by Santana:
- Starting from [tex]\((4, 0)\)[/tex],
- Reflecting across the y-axis to get [tex]\((-4, 0)\)[/tex],
- Then translating 2 units to the right to get [tex]\((-2, 0)\)[/tex],
is indeed correct. Santana correctly applies the transformations in the specified order. The final coordinates of the image are [tex]\((-2, 0)\)[/tex].
So, Santana's thinking is correct.
### Step 1: Reflection across the y-axis
1. Pre-image Coordinates: (4, 0)
- Santana starts with the point [tex]\((4, 0)\)[/tex], which is the pre-image.
2. Reflection across the y-axis:
- Reflecting a point [tex]\((x, y)\)[/tex] across the y-axis changes its coordinates to [tex]\((-x, y)\)[/tex].
- Applying this transformation to [tex]\((4, 0)\)[/tex]:
[tex]\[ (4, 0) \xrightarrow{\text{reflect across y-axis}} (-4, 0) \][/tex]
- This gives us the reflected coordinates [tex]\((-4, 0)\)[/tex].
### Step 2: Translation 2 units to the right
3. Translation 2 units to the right:
- Translating a point [tex]\((x, y)\)[/tex] 2 units to the right changes its coordinates to [tex]\((x + 2, y)\)[/tex].
- Applying this transformation to [tex]\((-4, 0)\)[/tex]:
[tex]\[ (-4, 0) \xrightarrow{\text{translate 2 units right}} (-2, 0) \][/tex]
- This gives us the final image coordinates [tex]\((-2, 0)\)[/tex].
### Verification of Santana’s Thinking
4. Initial Pre-image:
- The original pre-image coordinates are [tex]\((4, 0)\)[/tex].
5. Reflection Result:
- Reflecting [tex]\((4, 0)\)[/tex] across the y-axis yields [tex]\((-4, 0)\)[/tex].
6. Translation Result:
- Translating [tex]\((-4, 0)\)[/tex] 2 units to the right yields [tex]\((-2, 0)\)[/tex].
### Final Answer
The sequence of transformations suggested by Santana:
- Starting from [tex]\((4, 0)\)[/tex],
- Reflecting across the y-axis to get [tex]\((-4, 0)\)[/tex],
- Then translating 2 units to the right to get [tex]\((-2, 0)\)[/tex],
is indeed correct. Santana correctly applies the transformations in the specified order. The final coordinates of the image are [tex]\((-2, 0)\)[/tex].
So, Santana's thinking is correct.
