Problem 98 - College Algebra with Modeling and Visualization

Question

Suppose that the population of a species of fish (in thousands) is modeled by \(f(x)=\frac{x+10}{0.5 x^{2}+1},\) where \(x \geq 0\) is in years. (a) Graph \(f\) in \([0,12,1]\) by \([0,12,1] .\) What is the horizontal asymptote? (b) Determine the initial population. (c) What happens to the population of this fish? (d) Interpret the horizontal asymptote.

Step-by-Step Solution

Answer

The horizontal asymptote is \( y = 0 \). The initial population is 10,000 fish, and the population decreases over time, tending towards zero.

Step 1: Graph the Function

To determine the graph of the function \( f(x) = \frac{x+10}{0.5x^2+1} \) in the interval \([0,12,1]\), plot points by calculating the values of \( f(x) \) for \( x = 0, 1, 2, \ldots, 12 \). Use a graphing calculator or software for accuracy.

Step 2: Identify the Horizontal Asymptote

Horizontal asymptotes of rational functions are found by comparing the degrees of the polynomials in the numerator and denominator. For \( f(x) = \frac{x+10}{0.5x^2+1} \), the degree of the numerator is 1 and the degree of the denominator is 2. Since the denominator's degree is greater, the horizontal asymptote is \( y = 0 \).

Step 3: Determine the Initial Population

The initial population is found by evaluating \( f(x) \) at \( x = 0 \). Thus, \( f(0) = \frac{0+10}{0.5(0)^2+1} = 10 \). The initial population is 10,000 fish.

Step 4: Analyze the Population's Future Behavior

As \( x \) (years) increases, \( f(x) \) approaches its horizontal asymptote, \( y = 0 \). This indicates that the population decreases over time and tends towards zero.

Step 5: Interpret the Horizontal Asymptote

The horizontal asymptote \( y = 0 \) implies that as time goes to infinity, the population of fish approaches zero. Over a long period, without changes in the model's parameters, the species' population is expected to become extinct.

Key Concepts

Horizontal Asymptote

In population modeling, the concept of a horizontal asymptote is crucial for understanding the long-term behavior of a population. A horizontal asymptote of a function is a horizontal line that the graph of the function approaches as the input values become extremely large. When we deal with a rational function like \(f(x) = \frac{x+10}{0.5x^2+1}\), the horizontal asymptote can help us predict the eventual fate of a population. We determine the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials. In this case:

• The degree of the numerator \((x+10)\) is 1.
• The degree of the denominator \((0.5x^2+1)\) is 2.

Since the denominator's degree is greater than the numerator's, the horizontal asymptote is found at \(y = 0\). This indicates that as time goes on, the fish population will decrease towards zero, predicting a potential extinction of the species in the absence of other ecological factors.

Rational Functions

A rational function is a type of function represented by the fraction of two polynomials. They take the form \(f(x) = \frac{P(x)}{Q(x)}\) where \(P(x)\) and \(Q(x)\) are polynomial expressions, and \(Q(x) eq 0\). In our fish population model, the rational function \( f(x) = \frac{x+10}{0.5x^2+1} \) provides valuable insights into how the population changes over time.

• The numerator \((x+10)\) influences the growth initially as it determines instant rates of population increase.
• The denominator \((0.5x^2+1)\) dominates long-term behavior, showing how larger \(x\) values slow down growth, bringing the function closer to its asymptote.

Understanding these components allows us to interpret population dynamics effectively, considering initial growth phases and then predict the coming decline as dictated by the function.

Graphing Functions

Graphing functions is a fundamental skill in analyzing mathematical models, like those used in population studies. By graphing \(f(x) = \frac{x+10}{0.5x^2+1}\) in the interval \([0,12]\), we visualize how the fish population evolves over each year.To graph this function:

• Calculate \(f(x)\) for integers \(x = 0, 1, 2, \ldots, 12\).
• Use a graphing calculator or software for plotting the points precisely.
• Draw the curve that best fits the points to see the trend.

Graphing showcases the initial rise due to the numerator's influence and later, the decline as the denominator takes over, heading towards the horizontal asymptote \(y = 0\). This visual representation helps us comprehend the overall trajectory and potential endpoint of the population being studied.

Initial Population

The initial population provides the starting point of our analysis in population modeling and is essential for understanding the baseline from which any changes occur. It is the value of the function \(f(x)\) at \(x = 0\). For \(f(x) = \frac{x+10}{0.5x^2+1}\), calculating the initial population involves:

Substituting \(x=0\) into the function.

Evaluating \(f(0) = \frac{0+10}{0.5(0)^2+1} = 10\).

This computation tells us that the initial population is 10,000 fish. This figure serves as a benchmark against which future values are compared, helping to assess the function's projection's accuracy and understanding the implications of diminishing returns as years advance.