Let's analyze the transformation described by John and identify the mistake he made.
1. Understanding the Transformation Rule:
The given transformation rule is [tex]\( (x, y) \rightarrow (x + 4, y + 7) \)[/tex].
2. Applying the Transformation to the Pre-Image:
The pre-image coordinates provided are [tex]\( (4, 5) \)[/tex].
- For the [tex]\( x \)[/tex]-coordinate: [tex]\( x + 4 \)[/tex]
- Substituting [tex]\( x = 4 \)[/tex]:
[tex]\( x_{image} = 4 + 4 = 8 \)[/tex]
- For the [tex]\( y \)[/tex]-coordinate: [tex]\( y + 7 \)[/tex]
- Substituting [tex]\( y = 5 \)[/tex]:
[tex]\( y_{image} = 5 + 7 = 12 \)[/tex]
3. Resulting Image Coordinates:
After applying the transformation rule, the image coordinates should be [tex]\( (8, 12) \)[/tex].
4. Comparison with John's Claimed Result:
John claimed the image should be [tex]\( (0, -2) \)[/tex]. Let's see why this is incorrect.
- If we compare [tex]\( (8, 12) \)[/tex] with [tex]\( (0, -2) \)[/tex], we can see they are not the same.
- Thus, the transformation [tex]\( (x, y) \rightarrow (x + 4, y + 7) \)[/tex] does not map [tex]\( (4, 5) \)[/tex] to [tex]\( (0, -2) \)[/tex].
5. Identifying John's Error:
- The error lies in John's misunderstanding of the transformation rule.
- He likely miscalculated or assumed wrong values when applying [tex]\( (x + 4, y + 7) \)[/tex].
### Summary:
John's mistake was in the application of the transformation rule. When correctly applying [tex]\( (4, 5) \)[/tex] with the given rule [tex]\( (x + 4, y + 7) \)[/tex], we derive the image coordinates as [tex]\( (8, 12) \)[/tex], not [tex]\( (0, -2) \)[/tex].
