What Is a Stretch?

What the transformation keeps

Stretch is not a rigid motion because lengths and most angles do not stay the same. In many cases it is not even a similarity transformation, because different directions scale by different amounts.

This page keeps the original and stretched images together so you can study what remains aligned and what changes shape under nonuniform scaling.

A horizontal stretch by factor 2 doubles every x-coordinate relative to the chosen axis while leaving the y-coordinate unchanged. A square becomes a wider rectangle, which makes the loss of angle preservation easy to see. Naming the fixed direction first keeps the transformation from becoming a vague description of pulling.

A vertical stretch would do the same job in the other direction, changing heights while leaving horizontal positions alone. Comparing the two cases helps you see that the word stretch is incomplete until the direction and scale factor are named.

• A stretch changes one direction more than another.
• A one-way stretch has an invariant direction or line that acts as the reference for the change.
• Parallel relationships often survive a stretch, but lengths and angle measures generally do not.
• Under coordinate stretching, circles can become ellipses and squares can become rectangles or other distorted images.

Where stretches help

• Use stretch when studying graph transformations that change one axis more than the other.
• Use it in modelling and design contexts where a shape is elongated horizontally or vertically.
• Use it to compare rigid and non-rigid transformations on the same coordinate plane.

Stretch checks to make

• Do not call a stretch a dilation if the figure is not being scaled uniformly in all directions.
• Do not expect angle measures to stay fixed after a general stretch.
• Do not ignore which axis or direction is being stretched, because that decides the resulting shape.