S. D. V. Stephens

notes / Integrated Mathematics / chapter 8: College Algebra

28 Power Functions

A power function has the form .

Four graphs with the points at x=-1, 0, 1 marked: f(x)=x, a line; f(x)=x^2, a parabola; f(x)=x^3, an S-shaped curve with x^5 dashed beside it; and f(x)=x^4, a flat-bottomed U.

and show the shape for all odd ; and show the shape for all even . The blue dashed curve on the graph visualizes .

, odd , even
Key points , , , ,
Domain all real numbers all real numbers
Range or
Symmetry odd: symmetric about the origin even: symmetric about the -axis
Key points of x^n: for n odd, (-1,-1), (0,0) and (1,1); for n even, (-1,1), (0,0) and (1,1).

Remark. The higher in , the more dramatic the curve is (the more flat it becomes near the origin).

Example 28.1. Graph .

Show solution

Key points of , stretched by : Then the graph is shifted left and down , so the centre point lands at and the graph crosses the -axis at .

At : , giving the point .

Example 28.2. Graph .

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Key points of , reflected over the -axis, then shifted right and up :

Graphs of G(x)=3(x+2)^5-3, which flattens at (-2,-3) and crosses the x-axis at (-1,0), and F(x)=3-(x-2)^4, an upside-down flat-topped U with peak (2,3) through (1,2) and (3,2).

Example 28.3. Sketch , , and .

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Four quick sketches: f(x)=2x is a line through the origin; f(x)=-x^3 falls from upper left to lower right; f(x)=-3x^2 is a downward parabola; f(x)=7x^4 is a narrow upward U.