# What is the graph of a power function?

Nov 23, 2014

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 $\left(0 , 0\right)$ through $\left(1 , 1\right)$ and further.

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

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

5) $R = - 1$. Graph is a hyperbola going through point $\left(1 , 1\right)$ for $x = 1$. From this point it is diminishing to $0$, asymptotically approaching the X-axis for $x \rightarrow + \infty$. It is growing to $+ \infty$, asymptotically approaching the Y-axis for $x \rightarrow 0$.

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 $\left(0 , 0\right)$ 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.