Question

Point P(4,-2) undergoes a translation given by (x, y) - (x+3, x-a) , followed by another translation (x, y) - (x-b, x+7) to produce the image of P”(-5,-8). Find the values of a and b and point P’.

Answer 1

Assuming x - a = y - a and x + 7 = y + 7

Original Point P (4, -2)

Translated to Point P' (x + 3, y - a) = (4 + 3, -2 - a) = (7, -2 - a)

Translated to next point P'' = (x - b, y + 7) = (7 - b, -2 - a + 7) = (7 - b, 5 - a) = (-5, 8)

From the above changes, we can see that 7 - b = -5 and 5 - a = 8. Therefore:

`\begin{gathered} 7-b=-5 \\ 7+5=b \\ 12=b \end{gathered}`
`\begin{gathered} 5-a=8 \\ 5-8=a \\ -3=a \end{gathered}`

The value of a = -3 and b = 12.

The point P' (7, -2 - a) = (7, -2 - (-3)) = (7, 1). Point P' is at (7, 1).

To check if this is right, let's look at the original point again and its transformations.

P (4, -2) translated to (x + 3, y - a) = (4 + 3, -2 - (-3)) = (7, 1).

P' (7, 1) is then translated to ( x - b, y + 7) = (7 - 12, 1 + 7) = (-5, 8).

As mentioned in the question, P'' is indeed found at (-5, 8).