Problem 40 - Algebra 2

Question

For each function, find the percent increase or decrease that the function models. $$ y=0.8\left(\frac{1}{8}\right)^{x} $$

Step-by-Step Solution

Answer

The function represents a 87.5% decrease per period.

Step 1: Interpret the Given Function

In the given function, \(y=0.8\left(\frac{1}{8}\right)^{x}\), the base of the exponent is \(\frac{1}{8}\), which combines with the initial amount of '0.8' and represents the percent change per period (x).

Step 2: Find the Percent Change

The base of the exponential function being less than 1 indicates a decrease over time. We change it from decimal to percentage to find the rate of decrease. We subtract the decimal representation from 1 and multiply by 100%. That is, \( [1 - \frac{1}{8}] * 100% = 87.5% \). This indicates a decrease of 87.5% per period (x).

Step 3: Factor in the Initial Amount

The percent change found in the previous step is applied to the initial amount, which is 0.8 in this case. Thus, each step x will result in an amount that is 87.5% less than 0.8.

Key Concepts

Exponential Function

An exponential function is a type of mathematical expression in which the variable appears in the exponent. These functions involve constant multipliers, which translate into rapid increases or decreases. The general form of an exponential function is \( y = a \, b^x \), where:

• \(a\) is the initial or starting value, often referred to as the coefficient.
• \(b\) is the base of the exponent, representing the rate of change.
• \(x\) is the exponent and serves as the independent variable.

In exponential functions, the value of \(b\) determines the nature of the growth or decay. A value of \(b\) greater than 1 results in exponential growth, while a value between 0 and 1 indicates exponential decay. Thus, understanding the base is crucial to determining how the function behaves. In our example, \(y = 0.8\left(\frac{1}{8}\right)^{x}\), the base \(\frac{1}{8}\) suggests a decay process.

Exponential Decay

Exponential Decay describes a process where quantities decrease at rates proportional to their current value. It can be seen in various contexts such as physics, biology, and finance. The defining characteristic of exponential decay is the rate at which it diminishes, represented by a base between 0 and 1 in the exponential function.In real-world scenarios, exponential decay frequently models scenarios like radioactive decay, cooling processes, or depreciating assets. For our function, \(y = 0.8\left(\frac{1}{8}\right)^{x}\), the base of \(\frac{1}{8}\) signifies a substantial decrease each period. Specifically, this function demonstrates a decrease of 87.5% per period:

• We find this by calculating \(1 - \frac{1}{8} = \frac{7}{8}\), then converting this to a percentage: \(\frac{7}{8} \times 100\% = 87.5\%\).

This showcases exponential decay clearly by depicting how quickly values can shrink over consecutive intervals.

Mathematical Modelling

Mathematical modelling is the process of using mathematical structures and relationships to represent real-world situations. It serves as a powerful tool for understanding and predicting phenomena. By capturing essential details mathematically, it can provide insights and foresight into various fields such as economics, engineering, and environmental science.In the case of the function \(y = 0.8\left(\frac{1}{8}\right)^{x}\), we are modelling a scenario that involves a steep exponential decay rate. Mathematical models like this can simulate processes such as chemical reactions dropping off sharply as reactants are consumed, or technology rapidly losing value as new advancements emerge.

• It is crucial in problem-solving and decision-making processes, as it allows for quantitative analysis and predictions.
• By altering parameters like the base or initial value, different scenarios and assumptions can be tested easily.

Models simplify complex systems, making them manageable and comprehensible for analysis and discussion.