To analyze the end behavior of the function [tex]\( h(x) = 2(x-3)^2 \)[/tex], we observe what happens as [tex]\( x \)[/tex] approaches both negative infinity and positive infinity.
1. As [tex]\( x \)[/tex] approaches negative infinity:
- The term [tex]\( (x-3)^2 \)[/tex] grows very large because squaring any large negative or positive number results in a large positive number.
- Therefore, [tex]\( (x-3)^2 \)[/tex] will tend to positive infinity.
- Multiplying this term by 2 results in a value that also approaches positive infinity.
- Hence, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
2. As [tex]\( x \)[/tex] approaches positive infinity:
- Similar to the first case, [tex]\( (x-3)^2 \)[/tex] will again grow very large as [tex]\( x \)[/tex] moves towards positive infinity.
- This term will tend to positive infinity as well.
- Multiplying it by 2 still results in a value that approaches positive infinity.
- Hence, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
So, placing these correctly into the drop-down menu selections:
1. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
2. As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
