Problem 4 - College Algebra with Corequisite Support

Question

For the following exercises, identify whether the statement represents an exponential function. Explain. The average annual population increase of a pack of wolves is \(25 .\)

Step-by-Step Solution

Answer

The statement represents linear growth, not an exponential function.

Step 1: Understanding Exponential Functions

Exponential functions are equations of the form \(f(x) = a \, b^x\), where \(a\) is a constant, \(b\) is the base of the exponential, and \(x\) is the exponent or variable. The key characteristic of exponential functions is that they involve repeated multiplication and their growth rate is proportional to their size.

Step 2: Analyzing Given Information

The statement says that the average annual population increase of a pack of wolves is 25, which means every year the population increases by the same amount, 25 wolves. This indicates a constant increase.

Step 3: Difference Between Linear and Exponential Growth

Linear growth involves adding the same amount each year, as described in the given statement. Exponential growth, by contrast, involves multiplying by a consistent factor. We see that the statement describes a linear growth since the population grows by a constant addition of 25 wolves annually, instead of multiplying by a factor.

Step 4: Conclusion

Given that the growth described in the statement involves a constant number being added each year rather than being multiplied by a fixed rate, we can conclude that this is not an exponential function.

Key Concepts

Linear Growth vs Exponential Growth

When it comes to understanding different types of growth patterns, linear and exponential growth stand out as fundamental concepts. Linear growth occurs when a quantity increases by a fixed amount over regular intervals of time. For instance, if we add 25 wolves to a pack each year, the growth is linear because the same addition happens each year.
In contrast, exponential growth happens when a quantity grows by a constant percentage or factor over time. Imagine a situation where each year a wolf pack grows by multiplying its current size by a number, say by 10%. This percentage-based growth indicates exponential growth.

• In linear growth, addition is regular and constant.
• In exponential growth, multiplication is key.
• Linear growth follows a straight line while exponential growth curves upwards.

Understanding these differences can help us easily classify whether a situation represents linear or exponential growth.

Understanding Exponential Functions

Exponential functions form the backbone of modeling exponential growth scenarios. These functions are typically in the form of \( f(x) = a \, b^x \). For clarity:

• \(a\) is a constant that represents the starting amount or initial size of what is being measured.
• \(b\) is the base and indicates the growth factor.
• \(x\) serves as the exponent marking how time or periods affect growth.

A core feature of exponential functions is that they involve growth that is proportional to their current size.
Each year or time period, the amount grows by multiplying, making the growth curve steeper over time. That's why they start slowly but soon increase rapidly. Recognizing these traits helps in understanding why exponential functions are pivotal in analyzing scenarios like investments or population growth where compounding factors play a crucial role.

Population Growth Analysis

Analyzing population growth is vital in ecology and resource management. Two primary growth models are studied: linear and exponential. In a linear model, as with the pack of wolves increasing by 25 annually, the growth rate is steady and constant. Such a model is simple and predictable.
However, many real-world populations grow exponentially. This means their numbers increase faster as the population size grows, usually following the form \( P(t) = P_0 \, e^{rt} \), where:

• \(P(t)\) is the population size at time \(t\).
• \(P_0\) is the initial population size.
• \(r\) is the growth rate.
• \(t\) is time.

Exponential growth often cannot continue indefinitely due to environmental limitations, such as food or space, which lead to logistic growth curves eventually.
Therefore, analyzing whether growth is linear or exponential can have significant implications for planning and conservation efforts.