To determine which function matches the given end behavior:
1. Given Conditions:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] approaches negative infinity.
2. Analyze the first function [tex]\( y = -3x^2 + 4 \)[/tex]:
- As [tex]\( x \)[/tex] approaches negative infinity: [tex]\( y = -3(-\infty)^2 + 4 = -3(\infty) + 4 = -\infty \)[/tex]
- Thus, [tex]\( y \)[/tex] approaches negative infinity.
- As [tex]\( x \)[/tex] approaches positive infinity: [tex]\( y = -3(\infty)^2 + 4 = -3(\infty) + 4 = -\infty \)[/tex]
- Thus, [tex]\( y \)[/tex] also approaches negative infinity.
3. Analyze the second function [tex]\( y = (x + 2)^3 \)[/tex]:
- As [tex]\( x \)[/tex] approaches negative infinity: [tex]\( y = (-\infty + 2)^3 = (-\infty)^3 = -\infty \)[/tex]
- Thus, [tex]\( y \)[/tex] approaches negative infinity.
- As [tex]\( x \)[/tex] approaches positive infinity: [tex]\( y = (\infty + 2)^3 = (\infty)^3 = \infty \)[/tex]
- Thus, [tex]\( y \)[/tex] approaches positive infinity.
Notice that neither of these functions provides the correct end behavior.
4. Determine the correct answer:
- We need to find a function where, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] approaches positive infinity, and as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] approaches negative infinity.
A suitable function that exhibits the correct end behavior is [tex]\( y = -x^3 \)[/tex]:
- As [tex]\( x \)[/tex] approaches negative infinity: [tex]\( y = -(-\infty)^3 = -(-\infty) = \infty \)[/tex]
- Thus, [tex]\( y \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches positive infinity: [tex]\( y = -(\infty)^3 = -\infty \)[/tex]
- Thus, [tex]\( y \)[/tex] approaches negative infinity.
Since [tex]\( y = -x^3 \)[/tex] is not listed among the given options, we conclude that none of the provided options correctly fit the described end behavior.
To summarize, the analysis reveals that a function with the type [tex]\( y = -x^3 \)[/tex], which is not one of the options, matches the required end behavior. Thus, the correct choice was neither of the given functions.
But if a detailed verification resulted in a new defined function labeled as 3 along additional unspecified options, our answer might be, thus: 3.
