Question

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: Analyze x - coordinate change

To find the change in x - coordinate, we use the formula \(\Delta x=\text{image}_x-\text{pre - image}_x\). The pre - image x - coordinate is \(4\) and the image x - coordinate is \(0\). So \(\Delta x = 0 - 4=- 4\).

Step2: Analyze y - coordinate change

To find the change in y - coordinate, we use the formula \(\Delta y=\text{image}_y-\text{pre - image}_y\). The pre - image y - coordinate is \(5\) and the image y - coordinate is \(-2\). So \(\Delta y=-2 - 5=-7\).

Step3: Compare with John's rule

John's rule is \((x,y)\to(x + 4,y + 7)\), which implies a change of \(+4\) in x and \(+7\) in y. But from our calculations, the actual change is \(-4\) in x (i.e., \(x-4\)) and \(-7\) in y (i.e., \(y - 7\)). So John used addition instead of subtraction for both the x and y coordinates.

Answer:

John should have used subtraction (specifically \((x,y)\to(x - 4,y - 7)\)) instead of addition for the transformation rule, as the x - coordinate changes from \(4\) to \(0\) (a decrease of \(4\), so \(x-4\)) and the y - coordinate changes from \(5\) to \(-2\) (a decrease of \(7\), so \(y - 7\)), not an increase of \(4\) and \(7\) respectively.