Question

How does a dilation affect a figure on a coordinate plane?

Answer 1

Assuming the center of dilation is at point `{0,0}` on the coordinate plane and a factor of dilation `f`, a point `A{x,y}` will be transformed into point `A'{fx,fy}`.
See more details below.

Explanation:

Dilation or scaling is the transformation of the plane according to the following rules:

(a) There is a fixed point `O` on a plane or in space that is called the center of scaling.

(b) There is a real number `f!=0` that is called the factor of scaling.

(c) The transformation of any point `P` into its image `P'` is done by shifting its position along the line `OP` in such a way that the length of `OP'` equals to the length of `OP` multiplied by a factor `|f|`, that is `|OP'| = |f|*|OP|`. Since there are two candidates for point `P'` on both sides from center of scaling `O`, the position is chosen as follows: for `f>0` both `P` and `P'` are supposed to be on the same side from center `O`, otherwise, if `f<0`, they are supposed to be on opposite sides of center `O`.

It can be proven that the image of a straight line `l` is a straight line `l'`.
Segment `AB` is transformed into a segment `A'B'`, where `A'` is an image of point `A` and `B'` is an image of point `B`.
Dilation preserves parallelism among lines and angles between them.
The length of any segment `AB` changes according to the same rule above: `|A'B'| = f*|AB|`.

Using coordinates, the above properties can be expressed in the following form.
Assuming the center of dilation is at point `{0,0}` on the coordinate plane and a factor of dilation `f`, a point `A{x,y}` will be transformed into point `A'{fx,fy}`.
If the center of dilation is at point `C{p,q}`, the point `A{x,y}` will be transformed by dilation into `A'{p+f(x-p),q+f(y-q)}`

The above properties and other important details about transformation of scaling can be found on Unizor