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 no matter what n is. 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. You can then modify this as usual to get the graph of a coordinate transformation of a power function. In particular, a function of the form f(x) = axn is sometimes called the generalized power function with coefficient a and exponent n. If n is a whole number, then this is also called the monomial function with coefficient a and degree n. You get the graph of a generalized power function by taking the graph of the power function with the same exponent, and stretching it vertically by the coefficient. Then a polynomial function is a sum of monomial functions.
The permanent URI of this web page
is
https://tobybartels.name/MATH-1150/2024FA/powerfn/.