Answer :
To determine the end behavior of the function [tex]\( f(x) = 1536x + 30x^4 + 7680 - 60x^5 + 1440x^3 - 4800x^2 + 6x^6 \)[/tex], we need to analyze the dominant term in the polynomial, which is the term that will most influence the value of the function as [tex]\( x \)[/tex] becomes very large in magnitude (either positively or negatively).
The function can be rewritten as:
[tex]\[ f(x) = 6x^6 - 60x^5 + 30x^4 + 1440x^3 - 4800x^2 + 1536x + 7680 \][/tex]
To determine the end behavior:
1. Identify the leading term: The term with the highest power of [tex]\( x \)[/tex] is the leading term. In this case, the leading term is [tex]\( 6x^6 \)[/tex].
2. Analyze the leading term's behavior as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
- The leading term [tex]\( 6x^6 \)[/tex] dominates the behavior of the function.
- Since [tex]\( 6x^6 \)[/tex] is positive and grows very large as [tex]\( x \)[/tex] grows, [tex]\( y \)[/tex] will also go to [tex]\(\infty\)[/tex].
- As [tex]\( x \to -\infty \)[/tex]:
- The leading term [tex]\( 6x^6 \)[/tex] still dominates, but note that [tex]\( x^6 \)[/tex] for negative [tex]\( x \)[/tex] is also positive (since raising a negative number to an even power results in a positive number).
- Therefore, even as [tex]\( x \)[/tex] becomes very large and negative, [tex]\( 6x^6 \)[/tex] is positive and [tex]\( y \)[/tex] will go to [tex]\(\infty\)[/tex].
Summarizing the behavior:
- As [tex]\( x \to \infty \)[/tex]: [tex]\( y \to \infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex]: [tex]\( y \to \infty \)[/tex]
Thus, the correct answer is:
As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex] and as [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
The function can be rewritten as:
[tex]\[ f(x) = 6x^6 - 60x^5 + 30x^4 + 1440x^3 - 4800x^2 + 1536x + 7680 \][/tex]
To determine the end behavior:
1. Identify the leading term: The term with the highest power of [tex]\( x \)[/tex] is the leading term. In this case, the leading term is [tex]\( 6x^6 \)[/tex].
2. Analyze the leading term's behavior as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \to \infty \)[/tex]:
- The leading term [tex]\( 6x^6 \)[/tex] dominates the behavior of the function.
- Since [tex]\( 6x^6 \)[/tex] is positive and grows very large as [tex]\( x \)[/tex] grows, [tex]\( y \)[/tex] will also go to [tex]\(\infty\)[/tex].
- As [tex]\( x \to -\infty \)[/tex]:
- The leading term [tex]\( 6x^6 \)[/tex] still dominates, but note that [tex]\( x^6 \)[/tex] for negative [tex]\( x \)[/tex] is also positive (since raising a negative number to an even power results in a positive number).
- Therefore, even as [tex]\( x \)[/tex] becomes very large and negative, [tex]\( 6x^6 \)[/tex] is positive and [tex]\( y \)[/tex] will go to [tex]\(\infty\)[/tex].
Summarizing the behavior:
- As [tex]\( x \to \infty \)[/tex]: [tex]\( y \to \infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex]: [tex]\( y \to \infty \)[/tex]
Thus, the correct answer is:
As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex] and as [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
