Question

which of the following best describes the end behavior of the polynomial function $f(x) = -2x^4 + 5x^2 - 1$?

• as $x$ approaches infinity, $y$ oscillates.
• as $x$ approaches infinity, $y$ approaches negative infinity.
• as $x$ approaches infinity, $y$ approaches infinity.
• as $x$ approaches infinity, $y$ remains constant.

Explanation:

Step1: Identify Leading Term

The polynomial is \( f(x) = -2x^4 + 5x^2 - 1 \). The leading term is \( -2x^4 \) (highest degree term).

Step2: Analyze Leading Term Behavior

For a polynomial, end - behavior is determined by the leading term. The degree of the leading term (\(x^4\)) is even, and the leading coefficient (\(-2\)) is negative. When \(x\) approaches \(+\infty\) (or \(-\infty\) for even degree, the behavior at both ends is the same), we consider \(x^4\): as \(x\to\infty\), \(x^4\to\infty\). Then, multiplying by \(-2\) (a negative number), we get \(-2x^4\to -\infty\) as \(x\to\infty\). The other terms (\(5x^2\) and \(-1\)) become negligible compared to the leading term as \(x\) approaches infinity. So as \(x\) approaches infinity, \(y = f(x)\) approaches negative infinity.

Answer:

As \( x \) approaches infinity, \( y \) approaches negative infinity. (The option corresponding to this description)