Transformations of power functions with negative exponents

Functions: Fractional functions

Transformations of power functions with negative exponents

We have seen the shape of the graph of a power function #f(x)=\tfrac{1}{x^n}# with #n \gt 0# and #n# is an integer. Just as power functions of the form #y=x^n# with #n \gt 0# and #n# is an integer, we can transform these power functions.

TransformationsWe can transform the function #f(x)=\frac{1}{x^n}# in three different ways.

	Transformations	Examples
1	We shift the graph of #f(x)=\tfrac{1}{x^n}# upwards by #\green q#. The new function is \[f(x)=\frac{1}{x^n}+\green q\] Since the new function shifts upwards by #\green q#, the range becomes equal to #\ivoo{\green q}{\infty}# when #n# is even. When #n# is odd, the range becomes equal to all numbers except #\green q#. The domain stays the same. Hence, the horizontal asymptote is equal to #y=\green q# and the vertical asymptote remains equal to #x=0#.	shifting #f(x)=\frac{1}{x^4}# upwards by #\green3# gives #f(x)=\frac{1}{x^4}+\green3#
2	We shift the graph of #f(x)=\tfrac{1}{x^n}# to the right by #\blue p#. The new function becomes \[f(x)=\frac{1}{\left(x-\blue p\right)^n}\] Since the new function is shifted to the right by #\blue p#, the domain becomes equal to all numbers except #x=\blue p#. The range remains the same. Hence, the horizontal asymptote remains equal to #y=0# and the vertical asymptote becomes equal to #x=\blue p#.	shifting #f(x)=\frac{1}{x^3}# to the right by #\blue2# gives #f(x)=\frac{1}{\left(x-\blue2\right)^3}#
3	We multiply the graph of #f(x)=\tfrac{1}{x^n}# by #\purple a# relative to the #x#-axis. The new function becomes \[f(x)=\frac{\purple a}{x^n}\] When multiplying by a positive number, domain, range and asymptotes remain equal to the original function. On the other hand, if we multiply by a negative number, the function reverses. When #n# is odd, domain, range and asymptotes remain the same. That is not the case when #n# is even, there the range also reverses to #\ivoo{-\infty}{0}#. Domain and asymptotes remain the same.	multiplying #f(x)=\frac{1}{x^5}# by #\purple4# relative to the #x#-axis gives #f(x)=\frac{\purple4}{x^5}#

Multiple transformationsWe can also perform multiple transformation after one another.

For example: #f(x)=\frac{\purple3 }{\left(x-\blue4\right)^4}# is first a shift of #\blue4# to the right hand side and next a multiplication by #\purple3# relative to the #x#-axis, of the power function with negative exponent #f(x)=\frac{1}{x^4}#.

SymmetryWhen shifting to the right by #\blue p# we also saw a shift in the vertical asymptote. This means that the symmetry axis of an even function shifts to the right by #x=\blue p#.

A power function with an odd, negative exponent #n# has a symmetry point at #\rv{0,0}#. When shifting upwards by #\green q#, the symmetry point also shifts upwards by #\green q#. In the same manner when shifting to the right by #\blue p# the symmetry point also shifts to the right by #\blue p# to the point #\rv{\blue p, 0}#.

Take a look at the two graphs below.

The blue (solid) graph has formula:
\[y=-{{3}\over{x^6}}\]
What is the formula of the green (dashed) graph?

#y=# #-{{3}\over{\left(x+1\right)^6}}#

The point #\rv{1,-3}# lies on the blue graph, we will investigate where this same point is on the green graph. On the green graph this point lies at #\rv{0,-3}#.
Hence, the green graph is obtained from the blue graph by shifting the blue graph to the right by #1#.
Hence we replace all occurences of #x# in the formula for the blue graph #y=-{{3}\over{x^6}}# by #x+1#. This gives the following formula for the green graph:
\[y=-{{3}\over{\left(x+1\right)^6}}\]

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