Let's analyze the problem step-by-step.
1. Initial Point:
John starts with the pre-image at the point \((4, 5)\).
2. Transformation Rule:
The rule given for transformation is \((x, y) \rightarrow (x + 4, y + 7)\), which means you add 4 to the x-coordinate and 7 to the y-coordinate.
3. Applying the Transformation Rule:
Applying this transformation rule to the initial point \((4, 5)\) gives us:
[tex]\[
x' = 4 + 4 = 8
\][/tex]
[tex]\[
y' = 5 + 7 = 12
\][/tex]
So, the expected transformed point should be \((8, 12)\).
4. John's Transformed Point:
However, John claims that the resulting image after the transformation is the point \((0, -2)\).
5. Identifying the Error:
Now let's find where John went wrong by comparing the expected point \((8, 12)\) with the point \((0, -2)\) that he provided.
- For the x-coordinate:
[tex]\[
\text{Error}_x = 8 - 0 = 8
\][/tex]
- For the y-coordinate:
[tex]\[
\text{Error}_y = 12 - (-2) = 12 + 2 = 14
\][/tex]
John's error is that he incorrectly identified the point that the transformation rule translates [tex]\((4, 5)\)[/tex] to. The correct transformed point should be [tex]\((8, 12)\)[/tex], but he stated it was [tex]\((0, -2)\)[/tex], indicating errors of 8 units in the x-direction and 14 units in the y-direction.
