Power functions (§5.1)

For any real number n, the power function with exponent n is the function f such that f(x) = xⁿ for all possible x. Most of the examples from the library of functions in §3.4 of the textbook are power functions:

Identity function: n = 1;
Square function: n = 2;
Square-root function: n = ½;
Cube function: n = 3;
Cube-root function: n = ⅓;
Reciprocal function: n = −1.

We could also throw in the constant function with value 1; this is the power function with exponent 0.

You may have noticed some common features of the graphs of these functions. The point (1, 1) is on the graph of each of these functions, and that's no coincidence; (1, 1) is on the graph of every power function, because f(1) = 1ⁿ = 1. Most of these graphs also have (0, 0), but not the reciprocal function. This is because f(0) = 0ⁿ = 0 whenever n is positive, so (0, 0) is on the graph of every power function with a positive exponent. Another common point is (−1, −1); this is on the graph of the identity function, the cube function, the cube-root function, and the reciprocal function. This is because (−1)ⁿ = −1 whenever n is an odd integer, or more generally whenever n is a rational number with an odd numerator and an odd denominator (in lowest terms). There's one more useful point, even though it only appears on the graphs of two of the functions above (the square function and the constant function): (−1, 1). This appears on the graph of every power function whose exponent is an even integer, or more generally whose exponent is a rational number with an even numerator (in lowest terms).

To graph a power function, you can largely follow one of the graphs above with the same selection of these key points. Just keep in mind that the larger the exponent, the flatter the graph will be in the middle, and the steeper it will be on the sides. When the exponent is negative, a larger absolute value (so a smaller exponent) makes the graph steeper in the middle and flatter on the sides.

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