The *power function* is defined as #y = x^R#.

It has a domain of positive arguments #x# and is defined for all *real* powers #R#.

1) #R = 0#. Graph is a horizontal line parallel to the X-axis intersecting the Y-axis at coordinate #Y = 1#.

2) #R = 1#. Graph is a straight line going from point #(0,0)# through #(1,1)# and further.

3) #R > 1#. Graph grows from point #(0,0)# through point #(1,1)# to #+oo#, below the line #y = x# for #x in (0,1)# and then above it for #x in (1,+oo)#

4) #0 < R < 1#. Graph grows from point #(0,0)# through point #(1,1)# to #+oo#, above the line #y = x# for #x in (0,1)# and then below it for #x in (1,+oo)#

5) #R = -1#. Graph is a hyperbola going through point #(1,1)# for #x = 1#. From this point it is diminishing to #0#, asymptotically approaching the X-axis for #x rarr +oo#. It is growing to #+oo#, asymptotically approaching the Y-axis for #x rarr 0#.

6) #-1 < R < 0#. A hyperbola similar to the one for #R = -1# going below the graph of function #y=x^-1# for #x>1# and above it for #0 < x < 1#.

7) #R < -1#. A hyperbola similar to the one for #R = -1# going above the graph of function #y=x^-1# for #x>1# and below it for #0 < x < 1#.

The power function #y = x^R# with *natural* #R# can be defined for all real arguments #x#. It's graph for negative #x# will be symmetrical relative to the Y-axis to a graph for positive #x# if the power #R# is *even* or centrally symmetrical relative to the origin of coordinates #(0,0)# for *odd* power #R#.

*Negative integer* values of #R# can be used as a power for all non-zero arguments #x# with the same considerations of graph symmetry as above.

For more details please refer to Unizor lecture about the graph of a power function following the menu items *Algebra - Graphs - Power Function*.