Functions: Power functions
Transformations of power functions
We have seen the shape of the graph of a power function #f(x)=x^n# with integer #n \gt 0#. Just as we can transform the quadratic #y=x^2#, we can also transform power functions.
We can transform the function #f(x)=x^n# in three different ways.
| Transformations | Examples | |
| 1 |
We shift the graph of#f(x)=x^n# upwards by #\green q#. The new function is \[f(x)=x^n+\green q\] The vertex of an even power function and the symmetry point of an odd power function shifts upwards by #\green q#. For the new function the vertex, if an even power function, or the symmetry point, if an odd power function, becomes #\rv{0, \green q}#. |
shifting #f(x)=x^4# upwards by #\green3# gives #f(x)=x^4+\green3#
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| 2 |
We shift the graph of #f(x)=x^n# to the right by #\blue p#. The new function is \[f(x)=\left(x-\blue p\right)^n\] The vertex of an even power function and the symmetry point of an odd power function shifts to the right by #\blue p#. For the new function the vertex, if an even power function, or the symmetry point, if an odd power function, becomes #\rv{\blue p, 0}#. |
shifting #f(x)=x^3# to the right by #\blue2# gives #f(x)=\left(x-\blue2\right)^3#
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| 3 |
We multiply the graph of #f(x)=x^n# by #\purple a# relative to the #x#-axis. The new function is \[f(x)=\purple a x^n\] If #\purple a \lt 0# then the graph flips. If #\purple a =- 1#, then the new function is a reflection of the old function in the #x#-axis. |
multiplying #f(x)=x^5# by #\purple4# relative to the #x#-axis gives #f(x)=\purple4x^5#
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#y=# #3\cdot x^4-50#
The point #\rv{0,0}# lies on the blue graph, we will investigate where this same point lies on the green graph. On the green graph, this same lies at #\rv{0,-50}#.
Hence, the green graph is obtained by shifting the blue graph downwards by #50#.
We subtract #50# from the formula of the blue graph #y=3\cdot x^4#. This gives us the following formula for the green graph:
\[y=3\cdot x^4-50\]
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