The mathematical term for a function of a variable that changes with a fixed power (actually the sum of fixed multiples of fixed non-negative powers) is "polynomial." The multipliers are called "coefficients; the powers, "exponents." ("Fixed" here means constant, i.e, the values of the coefficients and the exponents don't change with the value of the variable.
The highest power, called the "degree" or "order" of the function, gives the name of the polynomial:
After exponent=5, the names are simply the "n-th power."
I'm not sure what you mean by "explain" the graph. A polynomial function operates by iterated multiplication on the value of the variable, the number of iterations given by its degree. Because multiplying a number between 0 and 1 by itself results in a product that's smaller than the number, the higher the degree of the polynomial, the slower its growth between 0 and 1.
In your graphical example, your polynomials have only one term each, and all their coefficients are 1. These are called "monic monomials." Their values are zero starting at 0 because multiplying 0 by itself any number of times results in 0. The values increase toward one as the variable approaches 1, reaching one at 1. This because multiplying 1 any number of times by itself just results in 1.