Understanding the End Behavior of f(x) = x5 + 8x4 + 16x3
To analyze the end behavior of the polynomial function f(x) = x5 + 8x4 + 16x3, we need to examine what happens to the value of f(x) as x approaches both positive and negative infinity.
General Rules for Polynomial Functions
The end behavior of a polynomial function is primarily determined by its leading term, which is the term with the highest degree. In our case, the leading term is x5. Since this is an odd-degree polynomial with a positive leading coefficient, we can make several observations:
As x approaches positive infinity (x → +∞)
As x increases towards positive infinity, the leading term x5 will dominate the function’s behavior. Thus, we can conclude that:
- f(x) → +∞ as x → +∞
As x approaches negative infinity (x → -∞)
Conversely, as x decreases towards negative infinity, the odd power of x in the leading term means that it will also become negative, effectively overriding the other terms. Therefore, we find that:
- f(x) → -∞ as x → -∞
Summary of End Behavior
In summary, the end behavior of the graph of f(x) = x5 + 8x4 + 16x3 can be described as follows:
- As x approaches positive infinity, f(x) approaches positive infinity.
- As x approaches negative infinity, f(x) approaches negative infinity.
This behavior indicates that the graph of the function will rise on the right side and fall on the left side, characteristic of many cubic functions.