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 $x \rightarrow-\infty, g(x) \rightarrow-\infty$.

Question

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 $x \rightarrow-\infty, g(x) \rightarrow-\infty$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine the end behavior of the polynomial function $g(x) = -x^4 + 2x^3 + 5x^2 - 1$, we consider the leading term. ‖ ‖ 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 $-\infty$.⚹◊What is the leading term of the polynomial function $g(x)=-x^{4}+2 x^{3}+5 x^{2}-1$ and how does it affect the end behavior of the graph of $g$ as $x$ approaches positive or negative infinity?⍭ Generate me a similar question◊

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