Question

Math

Posted 6 months ago

```
Consider the polynomial function $g(x)=-x^{4}+2 x^{3}+5 x^{2}-1$.
What is the end behavior of the graph of $g$ ?
Choose 1 answer:
(A) As $x \rightarrow \infty, g(x) \rightarrow \infty$, and as $x \rightarrow-\infty, g(x) \rightarrow \infty$.
(B) As $x \rightarrow \infty, g(x) \rightarrow-\infty$, and as $x \rightarrow-\infty, g(x) \rightarrow \infty$.
(C) As $x \rightarrow \infty, g(x) \rightarrow-\infty$, and as $x \rightarrow-\infty, g(x) \rightarrow-\infty$.
(D) As $x \rightarrow \infty, g(x) \rightarrow \infty$, and as $x \rightarrow-\infty, g(x) \rightarrow-\infty$.
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 2

The leading term of $g(x)$ is $-x^4$ since it has the highest power of $x$

step 3

As $x \rightarrow \infty$, the leading term $-x^4$ dominates the behavior of $g(x)$, causing $g(x) \rightarrow -\infty$

step 4

Similarly, as $x \rightarrow -\infty$, the leading term $-x^4$ still dominates, and since the power is even, $g(x) \rightarrow -\infty$

[question 1] Answer

C

Key Concept

End Behavior of Polynomial Functions

Explanation

The end behavior of a polynomial function is determined by its leading term. For even powers, the function approaches the same value for $x \rightarrow \infty$ and $x \rightarrow -\infty$. For negative leading coefficients, the function approaches $-\infty$.

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