1. The problem asks: What kind of polynomial functions are represented by the given expressions and graph description? 2. Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only non-negative integer powers of variables. 3. The given expressions involve powers of variables with integer exponents, including negative and fractional exponents, which are not allowed in polynomial functions. 4. Specifically, expressions like $a^{-n}$, $a^{-\frac{1}{n}}$, and fractions with variables in denominators indicate these are not polynomials but rather rational or radical expressions. 5. The graph described is a parabola opening upwards with a vertex near the origin, which is the typical shape of a quadratic polynomial function of the form $$y = ax^2 + bx + c$$ where $a \neq 0$. 6. Therefore, the polynomial functions related to the graph are quadratic polynomials, which are second-degree polynomials. 7. In summary, the algebraic expressions given are not polynomial functions due to negative and fractional exponents, but the graph corresponds to a quadratic polynomial function.