Problem 73 - A Graphical Approach to College Algebra

Question

Solve each problem. Suppose that an insect population in millions is modeled by $$f(x)=\frac{10 x+1}{x+1}$$ where \(x \geq 0\) is in months. (a) Graph \(f\) in the window \([0,14]\) by \([0,14] .\) Find the equation of the horizontal asymptote. (b) Determine the initial insect population. (c) What happens to the population after several months? (d) Interpret the horizontal asymptote.

Step-by-Step Solution

Answer

(a) Horizontal asymptote is \( y = 10 \). (b) Initial population is 1 million. (c) Population stabilizes at 10 million. (d) Asymptote \( y = 10 \) indicates long-term population.

Step 1: Plotting the Graph

To graph the function \( f(x) = \frac{10x+1}{x+1} \), we first note its behavior over the interval \([0, 14]\). As \( x \) approaches 0, the function behaves like a straight line (since \( 10x \) dominates), and as \( x \) becomes very large, the function approaches its horizontal asymptote. Plotting values from 0 to 14, you can manually find points or use a graphing tool to illustrate how \( f(x) \) behaves within the given window.

Step 2: Identify Horizontal Asymptote

To find the horizontal asymptote, consider the function as \( x \to \infty \). The horizontal asymptote reflects the limit \( \lim_{x \to \infty} f(x) = \lim_{x \to \infty} \frac{10x+1}{x+1} = \lim_{x \to \infty} \frac{10+\frac{1}{x}}{1+\frac{1}{x}} = 10 \). So, the horizontal asymptote is given by the equation \( y = 10 \).

Step 3: Calculate Initial Population

To find the initial population, substitute \( x = 0 \) into the function \( f(x) = \frac{10x+1}{x+1} \). This gives us the initial insect population: \( f(0) = \frac{10(0)+1}{0+1} = 1 \) million. Thus, the initial population is 1 million insects.

Step 4: Behavior of Population Over Time

As \( x \) increases indicating several months passing by, the function \( f(x) = \frac{10x+1}{x+1} \) approaches its horizontal asymptote, \( y = 10 \). This suggests that the insect population approaches 10 million as time progresses.

Step 5: Interpreting Horizontal Asymptote

The horizontal asymptote \( y = 10 \) means that, in the long term, the population stabilizes around 10 million. It represents the carrying capacity or a stable population number that it will reach eventually if there are no other changes in conditions.

Key Concepts

Horizontal Asymptote

A horizontal asymptote is a horizontal line that a graph approaches but never quite reaches. It indicates the behavior of a rational function as the input grows very large or very small. In the context of the function \(f(x) = \frac{10x+1}{x+1}\), the horizontal asymptote helps us understand the long-term behavior of the population model.

For the given function, we find the horizontal asymptote by taking the limit of \(f(x)\) as \(x\) approaches infinity. By simplifying, we get:

• Divide each term by \(x\) to give \( \lim_{x \to \infty} \frac{10 + \frac{1}{x}}{1 + \frac{1}{x}} \).
• As \(x\) becomes very large, \(\frac{1}{x}\) approaches 0.
• Thus, the limit simplifies to 10, making the horizontal asymptote \(y = 10\).

This asymptote indicates that as time progresses, the population approaches a stable value, suggesting a kind of equilibrium level that the population strives to achieve as months pass.

Initial Population

The initial population refers to the size of the population at the starting point of observation, which in this case is when \(x = 0\). To determine it, we simply substitute \(x = 0\) into the function:

• \(f(0) = \frac{10(0) + 1}{0 + 1} = 1\).

This calculation reveals that the initial population is 1 million insects.

Understanding the initial population provides a reference for measuring growth or decline over time. It tells us where the observed changes start and can help in predicting future questions or differences as months pass, based on the ongoing population growth through this model.

Graphing Functions

Graphing functions involves plotting points and examining the behavior of a function across its domain. For \(f(x) = \frac{10x+1}{x+1}\), graphing will visually demonstrate trends associated with the function. Let's break down the process:

• Start by calculating a few key values, especially at the beginning and end of the interval, like \(f(0)\) and close to the boundaries.
• Note how the graph approaches the horizontal asymptote \(y = 10\).
• Plot points at regular intervals within the given window of \([0, 14]\).

Graphing helps visualize how the population changes month by month up to the 14th month. It also highlights how the function behaves as \(x\) increases.

The overall shape of the graph will resemble an increasing curve that gradually flattens as it nears the asymptote, indicating a slowing growth rate as it stabilizes close to the asymptotic value. This overall view is vital for understanding the balance point between growth pressures and limiting factors impacting the population size.