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Power function

A power function is a function of the form $$y = \Tblue{a} x^{\Torange{n}}.$$ The number $n$ is the power. It is often an integer.

Power functions have special name depending on the value of the power.

 function power $\Torange{n}$ $2^{\Torange{n}}$ reciprocal constant linear quadratic cubic $-1$ $0$ $1$ $2$ $3$ $0.5$ $1$ $2$ $4$ $8$
Graph of the power functions $x^n$ for several values of $n$

A power function with a negative exponent is not defined for $x=0$.

A number raised to a negative power is the reciprocal of the number at the positive power. $$\Tblue{3}^{-\Torange{2}} = \frac{1}{\Tblue{3}^{\Torange{2}}} = \frac{1}{\Tgreen{9}}.$$

The graph of the sum of power functions is obtained by adding, for each $x$, the $y$-values of each individual function.

Graph of the sum of power function $y=x^{-2} - 2x.$ The graph of $y=x^{-2}$ is on the left, $y=-2x$ in the centre, and the sum on the right.