Problem 24 - Algebra 2

Question

Determine whether each function represents exponential growth or decay. $$ y=0.4\left(\frac{1}{3}\right)^{x} $$

Step-by-Step Solution

Answer

The function represents exponential decay.

Step 1: Identify the Function Structure

Observe that the function is given in the form \( y = a imes b^x \), where \( a = 0.4 \) and \( b = \frac{1}{3} \). This is a basic exponential function structure, suitable for identifying growth or decay.

Step 2: Analyze the Base (b)

In exponential functions, the base \( b \) determines the nature of the function. If \( b > 1 \), the function represents growth; if \( 0 < b < 1 \), the function represents decay. Here, \( b = \frac{1}{3} \), which is between 0 and 1.

Step 3: Conclude the Type of Exponential Function

Since the base \( b = \frac{1}{3} \) is between 0 and 1, the function \( y = 0.4 \left( \frac{1}{3} \right)^x \) represents exponential decay.

Key Concepts

Exponential Growth

Exponential growth happens when a quantity increases by a constant factor over equal intervals of time. In terms of algebra, an exponential growth function is written in the form \( y = a \times b^x \), where \( a \) is a constant, \( x \) is the variable, and the base \( b \) is greater than 1. The base value indicates how much the quantity multiplies for each increment of \( x \).
For example:

• If \( b = 2 \), it means the function doubles for each increase of \( x \) by 1.
• If \( a = 100 \) and \( b = 1.5 \), when \( x = 2 \), the function would be \( y = 100 \times 1.5^2 = 225 \), showing significant growth.

Exponential growth models are commonly used in real-world scenarios such as population growth, where a population grows rapidly over time, or in finance to model compound interest. The key takeaway is that the base of the exponential function being greater than 1 dictates the characteristic of growth, as it causes the value of \( y \) to escalate swiftly as \( x \) increases.

Exponential Decay

Exponential decay describes the process where a quantity decreases by a constant factor over equal time intervals. The typical form for exponential decay is \( y = a \times b^x \), where \( a \) is a constant, \( x \) is the variable, and \( b \) is the decay factor that lies between 0 and 1. The base value illustrates how much the quantity reduces for a unit increase in \( x \).
Consider this:

• For a base \( b = 0.5 \), the quantity halves every time \( x \) increases by one.
• If \( a = 200 \) and \( b = 0.6 \), when \( x = 3 \), the function appears as \( y = 200 \times 0.6^3 = 43.2 \), indicating a reduction over time.

Examples of exponential decay in real life include radioactive decay, where substances decrease in mass over time, or the depreciation of car value. The crucial detail to understand is sufficient: the base \( b \) being less than 1 is what causes the exponential function to decay, leading \( y \) to diminish as \( x \) rises.

Exponential Function Structure

The fundamental structure of an exponential function can be written as \( y = a \times b^x \), where \( a \) represents the initial amount, \( b \) denotes the base or multiplier, and \( x \) is the exponent or time variable. Understanding this structure is key to identifying and differentiating between exponential growth and decay.
Key elements include:

• \( a \): The initial value (when \( x = 0 \)) below which transformations take place.
• \( b \): The critical determinant of growth or decay.
  • If \( b > 1 \), it signifies exponential growth.
  • If \( 0 < b < 1 \), it signifies exponential decay.
• \( x \): Often represents time or intervals, influencing how the quantity transforms.

This foundational structure supports the analysis of numerous natural and human-made processes, furnishing predictions of outcomes over time. Recognizing whether \( b \) is greater than or less than 1 allows easy classification of the nature of the exponential relationship, providing insight into the behavior of complex systems.