Power Function with Natural Exponent

Function `y=x^n`, where n is natural number, is called power function with natural exponent.

power function with natural exponent

When n=1, we obtain function y=x; when n=2, we obtain parabola `y=x^2`; when n=3, we obtain cubic parabola `y=x^3`.

Let n is any even natural number greater than two: n=4, 6, 8, ... . In this case function `y=x^n` has same properties as `y=x^2`. Graph of such function resembles parabola, but for |x|>1 this function grows faster and appeares closer to y-axis; when |x|<1, the bigger n the closer graph to x-axis (see left figure).

Let n is any odd natural number greater than three: n=5, 7, 9, ... . In this case function `y=x^n` has same properties as `y=x^3`. Graph of such function resembles cubic parabola, but the bigger n the more steeply parts of graphs up and down. When |x|<1, the bigger n the closer graph to x-axis (see right figure).