To determine the end behavior of the function [tex]\( h(x) = 2(x-3)^2 \)[/tex], let's analyze how [tex]\( h(x) \)[/tex] behaves as [tex]\( x \)[/tex] approaches both negative infinity and positive infinity.
1. As [tex]\( x \)[/tex] approaches negative infinity:
Consider the term [tex]\( (x-3)^2 \)[/tex]. As [tex]\( x \)[/tex] becomes very large in the negative direction (i.e., as [tex]\( x \)[/tex] approaches negative infinity), the term [tex]\( (x-3) \)[/tex] will also be a large negative number. However, when this term is squared, it becomes a large positive number since squaring a number always results in a non-negative value.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( (x-3)^2 \)[/tex] becomes very large. Multiplying this large positive value by 2 does not change the fact that it is large and positive. Hence, when [tex]\( x \)[/tex] approaches negative infinity, the value of [tex]\( h(x) \)[/tex] will increase without bound.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
2. As [tex]\( x \)[/tex] approaches positive infinity:
Similarly, as [tex]\( x \)[/tex] becomes very large in the positive direction (i.e., as [tex]\( x \)[/tex] approaches positive infinity), the term [tex]\( (x-3) \)[/tex] will also be a large positive number. Squaring this positive number will result in a large positive value.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( (x-3)^2 \)[/tex] also becomes very large. Multiplying this large positive value by 2 similarly results in a larger positive value. Hence, as [tex]\( x \)[/tex] approaches positive infinity, the value of [tex]\( h(x) \)[/tex] will also increase without bound.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
Based on this analysis, the correct answers are:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( h(x) \)[/tex] approaches positive infinity.
