Problem 17 - College Algebra

Question

For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain. $$ y=11,701(0.97)^{t} $$

Step-by-Step Solution

Answer

The equation represents exponential decay since the base 0.97 is less than 1.

Step 1: Identify the Base of the Exponential Expression

The given equation is \( y = 11,701(0.97)^t \). In an exponential equation of the form \( a(b)^t \), the number \( b \) is called the base. Here, the base is \( 0.97 \).

Step 2: Analyze the Base Value

In any exponential function \( y = a(b)^t \), it represents exponential growth if \( b > 1 \) and exponential decay if \( 0 < b < 1 \). If the base is exactly \( 1 \), then it is neither growth nor decay. In this equation, the base \( 0.97 \) is less than 1.

Step 3: Conclude the Type of Exponential Equation

Since the base \( 0.97 \) is between 0 and 1, the equation \( y = 11,701(0.97)^t \) represents exponential decay. When the base is less than 1, the function value decreases as \( t \) increases, indicating decaying behavior.

Key Concepts

Exponential Growth

Exponential growth occurs when the value of a quantity increases at a rate proportional to its current value, meaning that as the quantity grows, so does the rate of growth. This is characterized by an exponential function of the form \( y = a(b)^t \), where \( a \) is a constant that represents the initial amount, \( b \) is the base of the exponential function, and \( t \) is the exponent that typically represents time.

For exponential growth:

• The base \( b \) must be greater than 1.
• The function describes rapid increases, leading the graph to rise sharply upwards as time progresses.

Exponential growth is frequently observed in natural phenomena such as population growth, compound interest in finance, and the spread of viruses. Understanding this concept helps in predicting scenarios where rapid growth occurs.

Base of Exponential Function

The base of an exponential function is a crucial component that determines the behavior of the function. In the equation \( y = a(b)^t \), the base is denoted by \( b \), and it significantly influences whether the function represents growth or decay.

Analyzing the base:

• If \( b > 1 \), the function represents exponential growth.
• If \( 0 < b < 1 \), the function signifies exponential decay.
• If \( b = 1 \), the function is a constant, with no growth or decay.

In the given exercise, the base of \( 0.97 \) was identified, meaning that the function represents decay, as \( 0.97 \) is less than 1. Understanding the base's value is essential for predicting how the function's value changes over time.

Exponential Function Analysis

Exponential function analysis involves evaluating the function's components to determine its growth or decay characteristics. The key is to analyze the equation’s form, identify the base, and assess its relation to 1.

When analyzing an exponential function:

• Identify the coefficient \( a \), which reveals the initial value when \( t = 0 \).
• Examine the base \( b \), since its relation to 1 (greater, less, or equal) decides the nature of the function.
• Interpret the exponent \( t \), often referring to time, to understand the change in the variable \( y \) over time.

In the exercise example, the base is \( 0.97 \), indicating exponential decay, as the value of \( y \) will decrease over time due to the base being less than 1. Mastery in analyzing exponential functions is fundamental for modelling real-life situations, like radioactive decay or depreciation of assets.