Final answer:
The end behavior of the polynomial function f(x) = -x^9 + bx^4 + c will have the graph approaching negative infinity as x approaches positive infinity and positive infinity as x approaches negative infinity, due to the negative leading coefficient and odd exponent.
Step-by-step explanation:
The student's question explores the end behavior of the polynomial function f(x) = -x^9 + bx^4 + c. The end behavior of a polynomial function is determined by the leading term, which is the term with the highest power of x. In this case, the leading term is -x^9. Because the leading coefficient (the coefficient of the leading term) is negative and the exponent is odd, the function will approach negative infinity as x approaches positive infinity, and it will approach positive infinity as x approaches negative infinity. The values of b and c do not affect the end behavior; they only affect the shape of the graph in between.