3.02 Power functions

Power functions

Functions with a form like $y=x^2$y=x2, $y=5x^3$y=5x3 and $y=ax^n$y=axn are known as power functions.

The following applet lets you see the general shape of power functions where $a=1$a=1 and the powers are positive integers.

• What happens at the extremities of the graph? Is it different for odd and even powers? 
• Also look carefully at the shape of the curves as they come close to the origin.
• Do you notice anything else about the curves?

Odd degree power functions all have the following properties:

• The graph has $180$180$^\circ$° rotational symmetry about the origin, that is $f(-x)=-f(x)$f(−x)=−f(x) for all $x$x, 
• The graph moves in opposite directions at the extremities
• They all pass through $\left(0,0\right)$(0,0), $\left(-1,-1\right)$(−1,−1) and $\left(1,1\right)$(1,1)
• Domain and range are both $\left(-\infty,\infty\right)$(−∞,∞)
• As $n$n increases, within the interval $\left[-1,1\right]$[−1,1] the graphs lie closer to the $x$x-axis, 'flatter' about the point of inflection
• As $n$n increases and $x$x extends beyond the interval $\left[-1,1\right]$[−1,1], the functions values grow further away from the $x$x-axis than those of lower powers    

Even degree power functions all have the following properties:

• The graph is symmetric about the $y$y-axis, that is $f(-x)=f(x)$f(−x)=f(x) for all $x$x 
• The graph moves in same directions at the extremities
• They all pass through $\left(0,0\right)$(0,0), $\left(-1,1\right)$(−1,1) and $\left(1,1\right)$(1,1)
• Domain is and $\left(-\infty,\infty\right)$(−∞,∞)and the range is$\left[0,\infty\right)$[0,∞)
• As $n$n increases, within the interval $\left[-1,1\right]$[−1,1] the graphs lie closer to the $x$x-axis, 'flatter' about the turning point
• As $n$n increases and $x$x extends beyond the interval $\left[-1,1\right]$[−1,1], the functions values grow further away from the $x$x-axis than those of lower powers