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#

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#

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#

#y=# #2\cdot x^6+100#
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{0,100}#.
Hence, the green graph is obtained by shifting the blue graph upwards by #100#.
We add #100# to the formula of the blue graph #y=2\cdot x^6#. This gives us the following formula for the green graph:
\[y=2\cdot x^6+100\]
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