Polynomial functions in context
Read the degree from the ends or from finite differences, the zeros from the x-intercepts, and the constant term as the starting value.
A worked example
y-values 1, 4, 11, 22, 37 at x = 0 to 4: first differences 3, 7, 11, 15; second differences 4, 4, 4. Constant second differences mean degree 2.
The mistake people actually make
In a regression model the constant is the value at the start, not a rate of change. Check the domain makes sense for the context.