Problem 19 - Algebra 2

Question

Without graphing, determine whether each function represents exponential growth or exponential decay. $$ y=0.8\left(\frac{1}{8}\right)^{x} $$

Step-by-Step Solution

Answer

The function \( y=0.8\left(\frac{1}{8}\right)^{x} \) represents exponential decay.

Step 1: Identify the Base

Firstly, find the base of the exponent in the function. In this problem, \( \frac{1}{8} \) is the base, which is present in the bracket.

Step 2: Compare the Base with 1

Once the base is identified, compare it with number 1. Since \( \frac{1}{8} \) is less than 1 and greater than 0, it signifies that the function undergoes exponential decay.

Step 3: Concluding the Result

Since the base is less than 1, the function \( y=0.8\left(\frac{1}{8}\right)^{x} \) represents exponential decay.

Key Concepts

Exponential Growth

Exponential growth occurs when the base of an exponential function is greater than 1. Imagine a situation where something keeps doubling every period, such as a population of rabbits. If the base of an exponential function is larger than 1, the function will demonstrate exponential growth. In mathematical terms, exponential growth is represented by a function of the form:

• \( y = a \, b^{x} \)
• Where \( a \) is the initial amount and \( b > 1 \) represents the growth factor.

In exponential growth, as time progresses, the value of \( y \) increases at a rapid rate. This type of growth is seen in various real-world examples like compound interest, population growth, and inflation. As the exponent \( x \) becomes larger, the value of the function grows dramatically, making it a very powerful model for growth scenarios. When working with exponential growth, it's essential to not only identify the base but also understand how it impacts the overall increase in the function's output.

Exponential Decay

Exponential decay takes place when the base of an exponential function is between 0 and 1. Unlike growth, decay represents a situation where the quantity decreases over time, shrinking consistently in each period. The function presented in the original exercise, \( y = 0.8\left(\frac{1}{8}\right)^{x} \), is an example of exponential decay.For a function to depict exponential decay, the setup follows:

• \( y = a \, b^{x} \)
• Where \( 0 < b < 1 \) indicates the decay factor.

An easy way to conceptualize this is thinking about a leaking bucket where water continues to flow out at a steady rate. In exponential decay, each unit of time sees a smaller fraction of the initial value remain. Common examples involve radioactive decay, depreciation in value of goods, and cooling of hot objects. Recognizing exponential decay involves identifying the base value and confirming that it is indeed less than 1 to signify that the function's output diminishes over time.

Base Comparison

Base comparison is integral in determining whether an exponential function signifies growth or decay. The base in the expression \( b \) serves as the crucial indicator.When evaluating an exponential function, always ask:

• If \( b > 1 \), then the function represents exponential growth.
• If \( 0 < b < 1 \), then the function represents exponential decay.

The importance of base comparison cannot be overstated. By accurately identifying and comparing the base to 1, you can deduce the behavior of the function without needing to graph it. This technique saves time and provides a clear understanding of how an expression behaves over time. For any given exponential function, looking at the base allows you to predict and explain the nature of the function: whether it grows by doubling, triples, or decays by halving, making it an invaluable tool in mathematics.