Answer :
To determine the end behaviors of the polynomial function [tex]\( f(x) = -2(x-2)^5 \)[/tex], we need to analyze its leading term. The leading term will dictate the end behavior as [tex]\( x \)[/tex] approaches positive and negative infinity.
### Step-by-Step Solution:
1. Identify the Leading Term:
Expand the polynomial [tex]\( f(x) = -2(x-2)^5 \)[/tex] to find the leading term. The leading term is derived from the highest power of [tex]\( x \)[/tex], which is the fifth power in this case:
[tex]\[ (x-2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32 \][/tex]
Since it is multiplied by [tex]\(-2\)[/tex], the leading term is:
[tex]\[ -2(x^5) = -2x^5 \][/tex]
2. Analyze the Leading Term:
The leading coefficient is [tex]\(-2\)[/tex] and the exponent is [tex]\(5\)[/tex] (which is odd).
3. Determine the End Behavior from the Leading Term:
- When the leading term has an odd exponent and a negative coefficient, the polynomial’s end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex]
Thus, for the function [tex]\( f(x) = -2(x-2)^5 \)[/tex]:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]), the function [tex]\( f(x) \)[/tex] decreases without bound ([tex]\( f(x) \to -\infty \)[/tex]).
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the function [tex]\( f(x) \)[/tex] increases without bound ([tex]\( f(x) \to \infty \)[/tex]).
### Conclusion:
Based on the analysis, the correct end behavior for [tex]\( f(x) = -2(x-2)^5 \)[/tex] is that the left end goes up and the right end goes down.
The correct answer is:
[tex]\[ \text{C. The left end goes up; the right end goes down.} \][/tex]
### Step-by-Step Solution:
1. Identify the Leading Term:
Expand the polynomial [tex]\( f(x) = -2(x-2)^5 \)[/tex] to find the leading term. The leading term is derived from the highest power of [tex]\( x \)[/tex], which is the fifth power in this case:
[tex]\[ (x-2)^5 = x^5 - 10x^4 + 40x^3 - 80x^2 + 80x - 32 \][/tex]
Since it is multiplied by [tex]\(-2\)[/tex], the leading term is:
[tex]\[ -2(x^5) = -2x^5 \][/tex]
2. Analyze the Leading Term:
The leading coefficient is [tex]\(-2\)[/tex] and the exponent is [tex]\(5\)[/tex] (which is odd).
3. Determine the End Behavior from the Leading Term:
- When the leading term has an odd exponent and a negative coefficient, the polynomial’s end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex]
Thus, for the function [tex]\( f(x) = -2(x-2)^5 \)[/tex]:
- As [tex]\( x \)[/tex] approaches positive infinity ([tex]\( x \to \infty \)[/tex]), the function [tex]\( f(x) \)[/tex] decreases without bound ([tex]\( f(x) \to -\infty \)[/tex]).
- As [tex]\( x \)[/tex] approaches negative infinity ([tex]\( x \to -\infty \)[/tex]), the function [tex]\( f(x) \)[/tex] increases without bound ([tex]\( f(x) \to \infty \)[/tex]).
### Conclusion:
Based on the analysis, the correct end behavior for [tex]\( f(x) = -2(x-2)^5 \)[/tex] is that the left end goes up and the right end goes down.
The correct answer is:
[tex]\[ \text{C. The left end goes up; the right end goes down.} \][/tex]
