Question

analyzing student work
john says the transformation rule (x, y) → (x + 4, y + 7)
can be used to describe the slide of the pre - image (4, 5)
to the image (0, −2). what was his error?

Explanation:

Step1: Apply John's Rule

Using the transformation rule \((x, y) \to (x + 4, y + 7)\) on the pre - image \((4,5)\), we substitute \(x = 4\) and \(y = 5\) into the rule.
For the \(x\) - coordinate: \(x+4=4 + 4=8\)
For the \(y\) - coordinate: \(y + 7=5+7 = 12\)
So according to John's rule, the image should be \((8,12)\), but the actual image is \((0,-2)\).

Step2: Find Correct Transformations

Let the correct transformation rule be \((x,y)\to(x + a,y + b)\), where \((x,y)=(4,5)\) and the image is \((0,-2)\).
For the \(x\) - coordinate: \(4+a=0\), so \(a=0 - 4=-4\)
For the \(y\) - coordinate: \(5 + b=-2\), so \(b=-2 - 5=-7\)
The correct transformation rule is \((x,y)\to(x-4,y - 7)\), which means John used addition instead of subtraction for both the \(x\) and \(y\) - coordinate transformations.

Answer:

John's error was that he used the transformation rule \((x,y)\to(x + 4,y + 7)\) (adding 4 to the \(x\) - coordinate and 7 to the \(y\) - coordinate) instead of the correct rule \((x,y)\to(x-4,y - 7)\) (subtracting 4 from the \(x\) - coordinate and 7 from the \(y\) - coordinate) to transform the pre - image \((4,5)\) to the image \((0,-2)\).