write a single translation in algebraic notation for the following
(x,y) (x-6,y+2) followed by (x,y)
(x+3,y+1)
the 2nd translation takes
(x-6,y+2) -> (x-6+3,y+2+1) = (x-3,y+3)
I read that as
(x,y) ---> (x-6,y+2)
followed by
(x,y) ---> (x+3, y + 1)
so:
(x,y) ---> (x-6,y+2) ---> (x-6 + 3,y+2 + 1)
or
(x,y) ----> (x-3, y+3)
To find the single translation in algebraic notation, we need to determine the transformation applied to the point (x, y) and express it as an equation.
For the first transformation, (x, y) → (x - 6, y + 2), we can observe that it involves a horizontal shift (x - 6) and a vertical shift (y + 2). Therefore, we can express this transformation as:
x' = x - 6
y' = y + 2
For the second transformation, (x, y) → (x + 3, y + 1), it also involves a horizontal shift (x + 3) and a vertical shift (y + 1). Thus, we can express this transformation as:
x'' = x + 3
y'' = y + 1
To find the single translation that combines both transformations, we can substitute the values of x' and y' into the second transformation equation. Hence, we get:
x'' = (x - 6) + 3
y'' = (y + 2) + 1
Simplifying these equations, we have:
x'' = x - 3
y'' = y + 3
Therefore, the single translation in algebraic notation for the given transformations is (x, y) → (x - 3, y + 3).