What is the end behavior of the graph of the polynomial function f(x) = x^5 – 9x^4 + 18x^3?

To understand the end behavior of the polynomial function f(x) = x⁵ – 9x⁴ + 18x³, we need to focus on its leading term, which is x⁵.

The leading term of a polynomial largely dictates its behavior as x approaches positive and negative infinity. In this case:

• As x → +∞ (x approaches positive infinity):
• The leading term x⁵ will dominate. Since x⁵ approaches +∞, we can conclude that:

f(x) → +∞

• As x → -∞ (x approaches negative infinity):
• The leading term x⁵ will also dominate. However, since we are dealing with an odd power, x⁵ approaches -∞. Thus, we have:

f(x) → -∞

In summary, the end behavior of the graph of the given polynomial function is:

• As x approaches positive infinity, f(x) approaches positive infinity.
• As x approaches negative infinity, f(x) approaches negative infinity.

This means that the graph will rise to the right and fall to the left, characteristic of polynomial functions with an odd degree and a positive leading coefficient.