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Consider the polynomial function h(x)=x^(6)-3x^(5)-17x^(3).
What is the end behavior of the graph of h ?
Choose 1 answer:
(A) As x rarr oo,h(x)rarr oo, and as x rarr-oo, h(x)rarr oo.
(B) As x rarr oo, h(x)rarr-oo, and as x rarr-oo,h(x)rarr oo.
(C) As x rarr oo, h(x)rarr-oo, and as x rarr-oo,h(x)rarr-oo.
(D) As x rarr oo,h(x)rarr oo, and as x rarr-oo, h(x)rarr-oo.

Consider the polynomial function h(x)=x63x517x3h(x)=x^{6}-3x^{5}-17x^{3}. \newlineWhat is the end behavior of the graph of hh?\newlineChoose 11 answer:\newline(A) As x,h(x)x \rightarrow \infty, h(x)\rightarrow \infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow \infty.\newline(B) As x,h(x)x \rightarrow \infty, h(x)\rightarrow -\infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow \infty.\newline(C) As x,h(x)x \rightarrow \infty, h(x)\rightarrow -\infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow -\infty.\newline(D) As x,h(x)x \rightarrow \infty, h(x)\rightarrow \infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow -\infty.

Full solution

Q. Consider the polynomial function h(x)=x63x517x3h(x)=x^{6}-3x^{5}-17x^{3}. \newlineWhat is the end behavior of the graph of hh?\newlineChoose 11 answer:\newline(A) As x,h(x)x \rightarrow \infty, h(x)\rightarrow \infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow \infty.\newline(B) As x,h(x)x \rightarrow \infty, h(x)\rightarrow -\infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow \infty.\newline(C) As x,h(x)x \rightarrow \infty, h(x)\rightarrow -\infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow -\infty.\newline(D) As x,h(x)x \rightarrow \infty, h(x)\rightarrow \infty, and as x,h(x)x \rightarrow -\infty, h(x)\rightarrow -\infty.
  1. Identify Leading Term: The problem asks us to determine the end behavior of the graph of the polynomial function h(x)=x63x517x3h(x) = x^6 - 3x^5 - 17x^3. To do this, we need to consider the leading term of the polynomial, as it dictates the end behavior of the graph.
  2. Analyze Leading Coefficient: The leading term of h(x)h(x) is x6x^6. The coefficient of this term is positive, and the degree is even. The end behavior of polynomial functions is determined by the leading term. For even-degree polynomials with a positive leading coefficient, as xx approaches infinity, the function approaches infinity, and as xx approaches negative infinity, the function also approaches infinity.
  3. Determine End Behavior: Therefore, based on the leading term x6x^6, we can conclude that as xx approaches infinity (xx \rightarrow \infty), h(x)h(x) approaches infinity (h(x)h(x) \rightarrow \infty), and as xx approaches negative infinity (xx \rightarrow -\infty), h(x)h(x) also approaches infinity (h(x)h(x) \rightarrow \infty). This matches option (A)(A) in the given choices.

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