This post assumes you already familiar with analyzing function translations. Even if you are, reading Function Transformations: Translation may be a useful introduction, as it uses this same approach to understanding transformations. Note that

– Translations move a graph, but do **not** change its shape

– Dilations change the shape of a graph, often causing “movement” in the process

The red curve in the image above is a “transformation” of the green one. It has been “dilated” (or stretched) horizontally by a factor of 3. A **dilation** is a stretching or shrinking about an axis caused by multiplication or division. You can think of a dilation as the result of drawing a graph on rubberized paper, stapling an axis in place, then either stretching the graph away from the axis in both directions, or squeezing it towards the axis from both sides.

Transformations are often easiest to analyze by focusing on how the location of specific points on the curve have changed. In the image above, the point on the green curve “corresponds” to point on the red curve. By this we mean that the transformation has moved point on the green graph to be at on the red graph.

### Horizontal Dilations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that the y-coordinate has not changed and the x-coordinate has been stretched to be three times further away from the y-axis… changing the shape of the curve in the process. If you were to examine any other pair of corresponding points, you would see the exact same scaling factor at work. This can be described algebraically by the equation:

Continue reading Function Transformations: Dilation