A dilation of the absolute value function alters the steepness of the "V" shape of its graph, making it either wider (compressed) or narrower (stretched).
Understanding Dilations
A dilation, in the context of the absolute value function, refers to a vertical stretch or compression. This transformation affects the slope of the two linear segments that form the characteristic "V" shape of the graph of y = |x|. The general form to represent a dilation is:
y = a|x|
where 'a' is the dilation factor.
Effect of the Dilation Factor 'a'
The value of 'a' determines the type of dilation:
-
|a| > 1 (Vertical Stretch): The graph becomes narrower (steeper). For example, y = 2|x| stretches the graph vertically, making the slopes of the lines forming the V steeper.
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0 < |a| < 1 (Vertical Compression): The graph becomes wider (less steep). For example, y = (1/3)|x| compresses the graph vertically, making the slopes of the lines forming the V less steep.
-
a < 0 (Reflection and Dilation): If 'a' is negative, the graph is reflected over the x-axis in addition to being stretched or compressed. For example, y = -2|x| would be a vertical stretch by a factor of 2 and a reflection over the x-axis, resulting in an upside-down, narrower V.
Example
Let's compare y = |x|, y = 2|x|, and y = (1/2)|x|.
| Function | Description | Effect on Graph |
|---|---|---|
| *y = | x | * |
| *y = 2 | x | * |
| *y = (1/2) | x | * |
In essence, the dilation factor 'a' acts like the slope in a linear equation (y = mx + b), controlling the steepness of the lines that make up the absolute value function's graph. A larger absolute value of 'a' results in a steeper, narrower V, while a smaller absolute value results in a shallower, wider V.
