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Sia

Question
Math
Posted 6 months ago

Consider the polynomial function g(x)=x4+2x3+5x21g(x)=-x^{4}+2 x^{3}+5 x^{2}-1.
What is the end behavior of the graph of gg ?

Choose 1 answer:
(A) As x,g(x)x \rightarrow \infty, g(x) \rightarrow \infty, and as x,g(x)x \rightarrow-\infty, g(x) \rightarrow \infty.
(B) As x,g(x)x \rightarrow \infty, g(x) \rightarrow-\infty, and as x,g(x)x \rightarrow-\infty, g(x) \rightarrow \infty.
(C) As x,g(x)x \rightarrow \infty, g(x) \rightarrow-\infty, and as x,g(x)x \rightarrow-\infty, g(x) \rightarrow-\infty.
(D) As x,g(x)x \rightarrow \infty, g(x) \rightarrow \infty, and as x,g(x)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)g(x) is x4-x^4 since it has the highest power of xx
step 3
As xx \rightarrow \infty, the leading term x4-x^4 dominates the behavior of g(x)g(x), causing g(x)g(x) \rightarrow -\infty
step 4
Similarly, as xx \rightarrow -\infty, the leading term x4-x^4 still dominates, and since the power is even, g(x)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 xx \rightarrow \infty and xx \rightarrow -\infty. For negative leading coefficients, the function approaches -\infty.

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