Transformations of root functions

Functions: Power functions and root functions

Transformations of root functions

Transformations

We can transform the function #f(x)=\sqrt{x}# in three different ways.

	Transformations	Examples
1	We shift the graph of #f(x)=\sqrt{x}# upwards by #\green q#. The new function becomes \[f(x)=\sqrt{x}+\green q\] Then the origin also shifts upwards by #\green q# and becomes equal #\rv{0, \green q}#. Therefore the range of the function becomes equal to #\ivco{\green q}{\infty}#. The domain does not change.	shifting #f(x)=\sqrt{x}# upwards by#\green3# gives #f(x)=\sqrt{x}+\green3#
2	We shift the graph of #f(x)=\sqrt{x}# to the right by #\blue p#. The new function becomes \[f(x)=\sqrt{x-\blue p}\] Then the origin also shifts to the right by #\blue p# and becomes equal to #\rv{\blue p, 0}#. Therefore the domain of the function becomes equal to #\ivco{\blue p}{\infty}#. The range does not change.	shifting #f(x)=\sqrt{x}# to the right by #\blue2# gives #f(x)=\sqrt{x-\blue 2}#
3	We multiply the graph of #f(x)=\sqrt{x}# by #\purple a# relative to the #x#-axis. The new function becomes \[f(x)=\purple a \sqrt{x}\] If #\purple a \gt 0#, the origin does not change. Domain and range also remain the same as with the old function. When multiplying by #\purple a \lt 0# the function reverses. The origin and the domain remain the same, but the range becomes equal to #\ivoc{-\infty}{0}#. If #\purple a =-1#, then we have a reflection across the #x#-axis of the old graph.	multiplying #f(x)=\sqrt{x}# by #\purple4# relative to the #x#-axis gives #y=\purple4 \sqrt{x}#

Multiple transformationsWe can also apply multiple transformations after one another.

For example: #f(x)=\purple3 \sqrt{x-\blue4}# first a shift of #\blue4# to the right and next a multiplication by #\purple3# relative to the #x#-axis of the root function #f(x)=\sqrt{x}#.

Take a look at the two graphs below.

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

#y=# #\sqrt{x}-1#

The blue graph has origin #\rv{0,0}#, we investigave where this same point is on the green graph. On the green graph this same point lies at #\rv{0,-1}#.
Hence, the green graph is obtained by shifting the blue graph downwards by #1#.
Hence, we subtract #1# from the formula of the blue graph #y=\sqrt{x}#. This gives the following formula for the green graph:
\[y=\sqrt{x}-1\]

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