Problem 31 - Algebra 2

Question

When \(a<0\) and \(b>1, y=a b^{x}\) models negative exponential growth. a. Open-Ended Write an exponential function that models negative growth. b. Give an example of a situation that could be modeled by your function. c. Critical Thinking Explain one difference between negative exponential growth and exponential decay.

Step-by-Step Solution

Answer

a) The exponential function representing negative growth is \(y=-2^x\) b) A situation modeled by this function could be a company whose profit has been halving every year. c) In negative exponential growth, the function becomes more negative as x increases while, in exponential decay, the values approach zero.

Step 1: Write Exponential Function Representing Negative Growth

Choose \(a = -1\) and \(b = 2\). So, the function is: \(y = -2^x\)

Step 2: Give an Example of Situation Modeled by Function

A company's profit has been halving every year for the past few years. This situation can be modeled by the function \(y = -2^x\), where x represents the number of years, and y is the profit of the company.

Step 3: Explain difference between Negative Exponential Growth and Exponential Decay

In negative exponential growth, the function decreases as x increases, but the values don't tend towards zero; they become more negative. In exponential decay, the function also decreases as x increases, but the values approach zero.

Key Concepts

Negative Exponential Growth

Negative exponential growth might sound unusual because when we hear "growth," we typically think of an increase in value. However, in mathematics, negative exponential growth refers to a situation where values become increasingly negative over time. This means that while the magnitude of the numbers grows, their actual values decrease even further into the negatives.

For example, if a scientist is tracking a chemical reaction where a substance's presence causes an increasingly greater reduction in temperature, this scenario could be modeled by a negative exponential growth function. In such a model, as time progresses on the x-axis, the temperature falls more steeply into negative values.

The general form of a negative exponential growth function is given by:

• \(y = a \cdot b^{x}\)

where \(a < 0\) and \(b > 1\). Here, \(x\) is typically the time variable, and \(y\) represents the outcome which becomes more negative. In practice, an example function could be \(y = -2^x\), whereby each increment in \(x\) leads to a larger negative \(y\).

Exponential Decay

Exponential decay is a process where quantities diminish over time, but unlike negative exponential growth, they approach zero. This concept is prevalent in natural processes and serves as a model for things like radioactive decay or cooling of hot objects in the environment.

An exponential decay function can be expressed as:

• \(y = a \cdot b^{x}\)

where \(a > 0\) and \(0 < b < 1\). Here, the base \(b\) is a fraction, indicating that with each step along the x-axis, the result is a smaller portion of the previous value.

Consider the case of a coffee left on a counter cooling down. Initially hot, the coffee gradually approaches room temperature over time; this cooling is typical of exponential decay where the temperature decreases rapidly at first, then slows as it nears the ambient temperature. The equation might look like \(y = 50 \cdot 0.5^x\), where \(y\) is the temperature and \(x\) is time.

Exponential Function Modeling

Exponential function modeling is a powerful tool in mathematics. It allows us to represent complex real-world phenomena with conciseness and clarity. By using exponential functions, we can predict behaviors, understand trends, and make informed decisions.

Modeling with exponential functions means choosing appropriate values for \(a\) and \(b\) in the equation \(y = a \cdot b^x\). Each choice affects the shape and direction of the curve. For instance, with \(a < 0\) and \(b > 1\), we model negative exponential growth, capturing scenarios like financial losses or increases in negativity.

Conversely, for exponential decay, we ensure \(a > 0\) and \(0 < b < 1\), which suit phenomena where diminishing returns or decreases occur over time, such as depreciation of an asset or radioactive substances losing half of their mass over set periods. Modeling accurately hinges on understanding the fundamental nature of the process being represented.