Graphs of Power Functions
For power functions with positive integer powers:
- A power function of the form is a parent function. The value of determines the shape of the graph.
- The graph of each parent function with a positive integer power contains the points and .
Transformations of Power Functions
Transformations can be used to make changes to the graph of a parent function. These include translations (shifts), stretches or compressions, and reflections.
For any function , the function can be translated vertically units and translated horizontally units. Horizontal translations are opposite in direction from the sign: Subtracting from the input translates the graph in the positive direction, and adding to the input translates it in the negative direction.
A function can also be stretched or compressed by a factor of and reflected across the -axis.
Vertical Translations | Horizontal Translations |
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For the graph of and :
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For the graph of and :
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Stretches and Compressions | Reflections |
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For the graph of and :
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For the graph of :
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Identify the parent function.
The first term in the polynomial is cubed.If the parent function is multiplied by , where , then the function is stretched. If , then the function is compressed. The value of in the given function indicates that the parent function is not stretched or compressed.