Question
Which of the following correctly describes the end behavior of the polynomial function, \( f(x)=-3 x^{4}+2 x^{2}-x \) ? A. The left end goes up and the right end goes down. B. Both ends go up. C. The left end goes down and the right end goes up. D. Both ends go down.
Ask by Vaughn Ball. in the United States
Mar 19,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
Both ends go down.
Solution
1. The given function is
\[
f(x) = -3x^4 + 2x^2 - x.
\]
2. The end behavior of a polynomial is determined by its leading term. In this function, the leading term is
\[
-3x^4.
\]
3. The degree of the polynomial is \(4\) (which is even), and the leading coefficient is \(-3\) (which is negative).
4. For an even-degree polynomial:
- If the leading coefficient is positive, both ends of the graph go up.
- If the leading coefficient is negative, both ends of the graph go down.
5. Since the leading coefficient here is negative, the graph of the polynomial will go down on both sides.
Therefore, the correct description is:
D. Both ends go down.
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Simplify this solution Bonus Knowledge
To determine the end behavior of the polynomial function \( f(x)=-3 x^{4}+2 x^{2}-x \), we focus on the leading term, which is \( -3x^4 \). Since the leading coefficient is negative and the degree (4) is even, both ends of the graph will point downwards. This means that as \( x \) approaches positive or negative infinity, \( f(x) \) will also approach negative infinity. Thus, the correct description is that both ends go down. So, the answer is D. Both ends go down.
