Question

1. describe the end behavior of the polynomial function (f(x)=-2x^{4}+2x^{3}+4x^{2}) using correct limit notation.

Explanation:

Step1: Identify the leading term

The leading term of \(f(x)=-2x^{4}+2x^{3}+4x^{2}\) is \(-2x^{4}\).

Step2: Analyze the degree and leading - coefficient

The degree \(n = 4\) (even) and the leading - coefficient \(a=-2\) (negative).

Step3: Find the limit as \(x\to+\infty\)

As \(x\to+\infty\), \(\lim_{x\to+\infty}f(x)=\lim_{x\to+\infty}(-2x^{4}+2x^{3}+4x^{2})\). Since the leading term dominates, \(\lim_{x\to+\infty}(-2x^{4}+2x^{3}+4x^{2})=\lim_{x\to+\infty}-2x^{4}=-\infty\) because for large positive \(x\), the term \(-2x^{4}\) is much larger in magnitude than the other terms and is negative.

Step4: Find the limit as \(x\to-\infty\)

As \(x\to-\infty\), \(\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}(-2x^{4}+2x^{3}+4x^{2})\). Since \((-x)^{4}=x^{4}\), the leading term \(-2x^{4}\) still dominates. So \(\lim_{x\to-\infty}(-2x^{4}+2x^{3}+4x^{2})=\lim_{x\to-\infty}-2x^{4}=-\infty\).

Answer:

\(\lim_{x\to+\infty}f(x)=-\infty\), \(\lim_{x\to-\infty}f(x)=-\infty\)