
A dilation is an enlargement (or reduction) that is "uniformly" applied to a figure. The image of a dilation is the same shape as the original figure, but is not necessarily the same size. Both the vertical length and horizontal length of a dilated figure are increased (or decreased) by the same factor. 
The dog on the right is a "uniform" enlargement, a dilation. 
But what do we have if a transformation only changes the vertical length of a figure (or only changes the horizontal length of the figure)?
What if only ONE direction is changed? If a figure is enlarged (or reduced) in only one direction, the change is referred to as a stretch.
In a stretch, the figure is distorted, and is not necessarily similar to the original figure. 
Horizontal Stretch
("Horizontal Dilation")
The width of the dog was increased, but the
height of the dog was NOT increased.
Only a horizontal change occurred. 

Stretches are defined in terms of a stretch factor and an invariant line. The invariant line acts as the reference location for the stretch, somewhat like the center of a dilation. Under a stretch, the perpendicular distance from an image point to the invariant line is the stretch factor times the perpendicular distance from the preimage point to the invariant line.
You have seen stretches applied to the graphs of functions. 
Vertical Stretch
("Vertical Dilation")
The height of the dog was increased, but the
width of the dog was NOT increased.
Only a vertical change occurred. 

Stretches on Coordinate Axis: 
We know that a dilation with a center at the origin and a scale factor of k
can be expressed as (x,y) → (kx, ky).
Notice that both the x and y coordinates are multiplied by the SAME value, k.
A stretch will expand the size of only ONE of the coordinates.
A stretch with stretch factor k and invariant xaxis: (x,y) → (x, ky).
The xcoordinate stayed the same
and the ycoordinate changed.
P' image of P with vertical stretch factor of 2
with invariant xaxis.

A stretch with stretch factor k and invariant yaxis: (x,y) → (kx, y).
The ycoordinate stayed the same
and the xcoordinate changed.
P' image of P with horizontal stretch factor of 2
with invariant yaxis
.

A combination of stretches:
A stretch with stretch factor a in the xdirection and stretch factor of b in the ydirection: (x,y) → (ax, by).
Point P was first horizontally stretched to (ax,y) and then was vertically stretched to P'(ax,by).



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