Complete each statement for the translation $T:\left(x,y\right)\to \left(x+3,y-1\right)$.

1. $T$ Glides points $\underset{¯}{?}$ units right and 1 unit $\underset{¯}{?}$.
2. The image of $\left(4,6\right)$ is $\left(\underset{¯}{?},\underset{¯}{?}\right)$.
3. The pre image of $\left(2,3\right)$ is $\left(\underset{¯}{?},\underset{¯}{?}\right)$.

Translation

In 2-dimensional geometry, a glide reflection is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation.

As per the given problem

Translation $T:(x,y)\to (x+3,y-1)$

Hence, x moves right 3 units & y moves 1 unit down.

Translation $T:(x,y)\to (x+3,y-1)$

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it. The original object is called the pre-image, and the translation is called the image.

As per the given problem

Translation $T:(x,y)\to (x+3,y-1)$

$T(4,6)\to (4+3,6-1)=(7,5)$

Hence, the image of $(4,6)$ is $(7,5)$.

Translation $T:(x,y)\to (x+3,y-1)$

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it. The original object is called the pre-image, and the translation is called the image.

As per the given problem

Translation $T:(x,y)\to (x+3,y-1)$

$T^{-1}(x,y)\to (x-3,y+1)$

$T^{-1}(2,3)\to (2-3,3+1)=(-1,4)$

Hence, the preimage of $(2,3)$ is $(-1,4)$.