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.
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:
Two earlier posts provide background information for this one: Function Translations and Function Dilations. If you are not already familiar with these topics, you may benefit from reading those first.
Given two points on a curve and their corresponding points after transformation, how does one determine the underlying transformations? Since two dilations and two translations may be taking place, it can be complex to try to separate the effects of dilation from those of translation.
As an example, consider the two curves above. The green curve is the graph of
and the red curve is a transformation of the green one. Two points are labeled on the green curve:
and their corresponding transformed points are labeled on the red curve:
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A function has been “dilated” (note the spelling… it is not spelled or pronounced “dialated”) when it has been stretched away from an axis or compressed toward an axis.
Imagine a graph that has been drawn on elastic graph paper, and fastened to a solid surface along one of the axes. Now grasp the elastic paper with both hands, one hand on each side of the axis that is fixed to the surface, and pull both sides of the paper away from the axis. Doing so “dilates” the graph, causing all points to move away from the axis to a multiple of their original distance from the axis. As an example of this, consider the following graph:
The graph above shows a function before and after a vertical dilation. The coordinates of two points on the solid line are shown, as are the coordinates of the two corresponding points on the dashed line, to help you verify that the dashed line is exactly twice as far from the x-axis as the same color point on the solid line.