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.
