Question

Math

Posted 3 months ago

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Q4.7 End Behavior of Polynomials II
1 Point
Determine the end behavior of the graph of $f(x)=-5 x^{4}-x^{3}+3 x^{2}+7$ : as $x \rightarrow \infty, f(x) \rightarrow \ldots$; $(0.5 \mathrm{pt})$
$\infty$
$-\infty$
as $x \rightarrow-\infty, f(x) \rightarrow$ $\qquad$ . $(0.5 \mathrm{pt})$
$\infty$
$-\infty$
```

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Answer from Sia

Posted 3 months ago

Solution by Steps

step 1

Identify the leading term of the polynomial $f(x) = -5x^4 - x^3 + 3x^2 + 7$. The leading term is $-5x^4$

step 2

Determine the end behavior as $x \rightarrow \infty$. Since the leading term $-5x^4$ has a negative coefficient and an even exponent, as $x \rightarrow \infty$, $f(x) \rightarrow -\infty$

step 3

Determine the end behavior as $x \rightarrow -\infty$. Similarly, since the leading term $-5x^4$ has a negative coefficient and an even exponent, as $x \rightarrow -\infty$, $f(x) \rightarrow -\infty$

Answer

As $x \rightarrow \infty, f(x) \rightarrow -\infty$

As $x \rightarrow -\infty, f(x) \rightarrow -\infty$

Key Concept

End Behavior of Polynomials

Explanation

The end behavior of a polynomial function is determined by its leading term. For a polynomial with a negative leading coefficient and an even exponent, the function will approach $-\infty$ as $x$ approaches both $\infty$ and $-\infty$.

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