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