What is the end behavior of the graph of the polynomial function [tex]f(x)=3 x^0+30 x^3+75 x^4[/tex]?
A. As [tex]x \rightarrow-\infty, y \rightarrow-\infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow-\infty[/tex].
B. As [tex]x \rightarrow-\infty, y \rightarrow-\infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow \infty[/tex].
C. As [tex]x \rightarrow-\infty, y \rightarrow \infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow-\infty[/tex].
D. As [tex]x \rightarrow-\infty, y \rightarrow \infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow \infty[/tex].
Answer (2)
The end behavior of the polynomial function f ( x ) = 3 + 30 x 3 + 75 x 4 indicates that as x approaches both positive and negative infinity, f ( x ) approaches positive infinity. Therefore, the correct option is D: As x \rightarrow -\infty, y \rightarrow +\infty \text{ and as } x \rightarrow +\infty, y \rightarrow +\infty.
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Identify the leading term: 75 x 4 .
Determine the degree: 4 (even) and the leading coefficient: 75 (positive).
As x approaches − ∞ , y approaches ∞ .
As x approaches ∞ , y approaches ∞ . The end behavior is: As x → − ∞ , y → ∞ and as x → ∞ , y → ∞
Explanation
Simplify the Polynomial We are given the polynomial function f ( x ) = 3 x 0 + 30 x 3 + 75 x 4 . First, we simplify the function by noting that x 0 = 1 , so f ( x ) = 3 + 30 x 3 + 75 x 4 . We want to determine the end behavior of the graph of this polynomial.
Identify Leading Term and Degree The end behavior of a polynomial is determined by its leading term, which is the term with the highest degree. In this case, the leading term is 75 x 4 . The degree of the polynomial is 4, which is an even number. The leading coefficient is 75, which is a positive number.
Determine End Behavior Since the degree is even and the leading coefficient is positive, as x approaches positive or negative infinity, the function f ( x ) approaches positive infinity. In other words, as x \tends − ∞ , y \tends ∞ , and as x \tends ∞ , y \tends ∞ .
Examples
Understanding the end behavior of polynomial functions is useful in many real-world applications. For example, when modeling population growth or decline, the end behavior of the polynomial model can tell us whether the population will eventually grow without bound or decline to zero. Similarly, in engineering, understanding the end behavior of a polynomial function can help predict the long-term stability of a system.
