If the leading coefficient is negative, the polynomial function will eventually decrease to negative infinity; if the leading coefficient is positive, the polynomial function will eventually increase to positive infinity.
This answer assumes by leading coefficient you mean the coefficient of the highest powered ##x## term (the normal usage).
If ##g(x)## is a polynomial with greatest degree ##n##
then if ##m > n##, the absolute value of ##x^m## will be greater than the absolute value of ##g(x)## once ##x## becomes sufficiently large.
For example if ##g(x) = 5x^2 – 4x +12##
##x^3## will have an absolute value greater than ##g(x)## provided x is greater than 3.
Since the term with the highest exponent will eventually provide a value that exceeds the combined value of all other terms, the sign of that term determines the eventual direction of the function value.
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