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Answer :
To determine the end behavior of the function [tex]\( f(x) = 3072 - 6x^5 + 78x^4 - 1680x^2 + 1536x - 60x^3 \)[/tex], we focus on the leading term of the polynomial. The leading term is the term with the highest power of [tex]\( x \)[/tex], which in this case is [tex]\(-6x^5\)[/tex].
Here's how we analyze the end behavior:
1. Identify the Leading Term: The leading term of the polynomial is [tex]\(-6x^5\)[/tex]. This term will dominate the behavior of the polynomial as [tex]\( x \)[/tex] becomes very large or very small.
2. Analyze the Coefficient: The coefficient of the leading term is [tex]\(-6\)[/tex], which is negative. The power [tex]\( 5 \)[/tex] is an odd number.
3. Determine the End Behavior:
- As [tex]\( x \rightarrow \infty \)[/tex] (as [tex]\( x \)[/tex] increases without bound), the expression [tex]\(-6x^5\)[/tex] becomes very large but negative because the coefficient is negative and the power is odd. Thus, [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex] (as [tex]\( x \)[/tex] decreases without bound), the expression [tex]\(-6x^5\)[/tex] becomes very large but positive. This is because an odd power of a negative number remains negative, but the negative coefficient [tex]\(-6\)[/tex] makes the result positive. Thus, [tex]\( f(x) \rightarrow \infty \)[/tex].
Therefore, the end behavior of the function is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Here's how we analyze the end behavior:
1. Identify the Leading Term: The leading term of the polynomial is [tex]\(-6x^5\)[/tex]. This term will dominate the behavior of the polynomial as [tex]\( x \)[/tex] becomes very large or very small.
2. Analyze the Coefficient: The coefficient of the leading term is [tex]\(-6\)[/tex], which is negative. The power [tex]\( 5 \)[/tex] is an odd number.
3. Determine the End Behavior:
- As [tex]\( x \rightarrow \infty \)[/tex] (as [tex]\( x \)[/tex] increases without bound), the expression [tex]\(-6x^5\)[/tex] becomes very large but negative because the coefficient is negative and the power is odd. Thus, [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex] (as [tex]\( x \)[/tex] decreases without bound), the expression [tex]\(-6x^5\)[/tex] becomes very large but positive. This is because an odd power of a negative number remains negative, but the negative coefficient [tex]\(-6\)[/tex] makes the result positive. Thus, [tex]\( f(x) \rightarrow \infty \)[/tex].
Therefore, the end behavior of the function is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
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