91Ó°ÊÓ

The initial population refers to the number of individuals at the very beginning of our observation, which is usually when time, denoted as \( t \), is zero. For the given problem, the population function is defined by the equation:
\[ y = \frac{6t + 12}{2t + 1} \]
To find the initial population, we substitute \( t = 0 \) into the function. This allows us to determine what the population quantity is when time has not yet started passing:
- Substitute \( t = 0 \) into the formula: \[ y(0) = \frac{6(0) + 12}{2(0) + 1} = \frac{12}{1} = 12 \]
From this process, it becomes clear that the initial population is 12 thousand bacteria. This number gives us a starting point, from which we can observe how the population changes overtime under the given conditions.

To understand the long-term population, we look at the behavior of the function as time goes to infinity (\( t \to \infty \)). This involves finding what is known as the horizontal asymptote of the function. A horizontal asymptote helps us predict the population when enough time has passed so that transient dynamics have settled.
For the function \( y(t) = \frac{6t + 12}{2t + 1} \), as \( t \) becomes very large, the behavior is dominated by the highest degree terms in both the numerator and the denominator:

• The numerator's highest degree term is \( 6t \).
• The denominator's highest degree term is \( 2t \).

When simplifying the function for large \( t \), we find:
\[ y(t) \approx \frac{6t}{2t} = 3 \]
Thus, as \( t \to \infty \), the long-term population approaches 3 thousand bacteria. This asymptotic analysis tells us that over a long period, the bacterial population will stabilize at this lower figure compared to its initial state.

A graphing utility can visually represent the changes in population by plotting the function \( y(t) = \frac{6t + 12}{2t + 1} \) over a suitable range of time \( t \). This practical tool helps verify theoretical calculations and provides a clear picture of how the population evolves over time.
When using a graphing utility, input the function and set a range for \( t \). As the graph is drawn, several key details emerge:

• At \( t = 0 \), the population starts at 12, confirming our initial population finding.
• As \( t \) increases, the curve approaching a horizontal line at \( y = 3 \) indicates the long-term population.
• The trend shows a decreasing function, from 12 to 3, matching our calculations regarding the population’s decline over time.

Graphing functions like this one make abstract concepts more concrete and allow for a better understanding of trends and behaviors in mathematical models.