The end behavior of a polynomial function is determined primarily by its leading term, which is the term with the highest degree. In the case of the polynomial function f(x) = 2x3 + 26x + 24, the leading term is 2x3.
Since the degree of the leading term is 3 (an odd number) and the leading coefficient (2) is positive, we can predict how the function will behave as x approaches positive and negative infinity.
As x approaches positive infinity:
When x becomes very large (i.e., x > 0), the leading term 2x3 will dominate the behavior of the function. Therefore, as x approaches positive infinity, f(x) will also approach positive infinity:
lim(x→∞) f(x) = ∞As x approaches negative infinity:
Conversely, when x becomes very negative (i.e., x < 0), the leading term 2x3 will still dictate the behavior, but since the cubic term remains negative (as raising a negative number to an odd power results in a negative), we find that as x approaches negative infinity, f(x) will approach negative infinity:
lim(x→−∞) f(x) = −∞In summary:
- As x → ∞, f(x) → ∞
- As x → −∞, f(x) → −∞
This analysis of the end behavior gives us a clear understanding of how the graph of the function behaves at the extremes, helping in painting the overall picture of its shape and direction.