Question

sec 5.3 graphs of polynomial functions: probl
(1 point)
determine the end behavior for ( y = - x ^ { 5 } + 5 x ^ { 4 } + 8 x ).

• as ( x

ightarrow - infty , y
ightarrow ) help (numbers)

• as ( x

ightarrow infty , y
ightarrow ) help (numbers)
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page generated october 28, 2025, 1:09:20 pm cdt

Explanation:

Step1: Identify the leading term

The leading term of the polynomial \(y = -x^{5}+5x^{4}+8x\) is \(-x^{5}\). The degree \(n = 5\) (odd) and the leading coefficient \(a=- 1\) (negative).

Step2: Apply the end - behavior rules

For a polynomial \(y = a x^{n}+\cdots\), when \(n\) is odd:

• If \(a>0\), as \(x

ightarrow-\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow\infty\), \(y
ightarrow\infty\).

• If \(a < 0\), as \(x

ightarrow-\infty\), \(y
ightarrow\infty\) (because \((-x)^{n}\) with odd \(n\) is \(-x^{n}\), and \(a(-x)^{n}\) when \(a=-1\) gives \((-1)\times(-x^{n})=x^{n}\) for large \(|x|\)) and as \(x
ightarrow\infty\), \(y
ightarrow-\infty\) (since \(a x^{n}\) with \(a=-1\) and odd \(n\) is \(-x^{n}\)).

Answer:

As \(x
ightarrow-\infty\), \(y
ightarrow\infty\)
As \(x
ightarrow\infty\), \(y
ightarrow-\infty\)