Problem 12 - Algebra 1: Concepts and Skills

Question

Identify the initial amount and the decay factor in the exponential function. $$ y=10(0.2)^{t} $$

Step-by-Step Solution

Answer

The initial amount 'a' is 10 and the decay factor 'b' is 0.2.

Step 1: Identify the Initial Amount

Looking at the standard form of an exponential function \(y=ab^{t}\) and comparing with the given equation \(y=10(0.2)^{t}\), we can identify 'a' as 10, since it is the number that the decay factor is being multiplied by.

Step 2: Identify the Decay Factor

The decay factor 'b' is the base of the exponent in the function. Here, 'b' can be identified as 0.2, since it is the base of the exponent.

Key Concepts

Initial Amount

The initial amount in an exponential function is a crucial aspect to understand. It represents the starting point or the original value before any changes occur over time. In our given function, which is expressed as \(y = 10(0.2)^t\), the initial amount is represented by the number that stands alone, before the parenthesis. In this scenario, it is \(10\). This means that at time \(t = 0\), or at the beginning of the observation, the quantity being measured begins at a value of \(10\).

• The initial amount can be thought of as the value at which the exponential process begins.
• It's independent of time and only reflects the starting condition of the function.
• In mathematical terms, it is symbolized by \(a\) in the standard form of an exponential function.

Understanding the initial amount helps clarify the baseline upon which growth or decay processes act, making it foundational for understanding and applying exponential functions effectively.

Decay Factor

The decay factor in an exponential function indicates how quickly a quantity decreases over time. It's crucial for modeling situations where something diminishes multiplicatively rather than linearly, such as radioactive decay or cooling processes. In the exponential function \(y = 10(0.2)^t\), the decay factor is \(0.2\), which is the base of the exponent. This number plays a significant role in determining how the value of \(y\) changes as time \(t\) increases.

• A decay factor less than \(1\) indicates that the function is decreasing over time.
• In our example, every time \(t\) goes up by one unit, the quantity is multiplied by \(0.2\), reducing its size.
• This factor is constant; it doesn’t change with time but acts uniformly throughout the process.

In essence, the decay factor provides insight into the rate at which the initial quantity will reduce, laying the groundwork for predicting future values.

Standard Form of Exponential Function

The standard form of an exponential function is foundational for understanding many real-world processes. It is generally represented as \(y = ab^t\). Each part of this equation serves a purpose:

• \(a\) represents the initial amount, or the starting value at \(t = 0\).
• \(b\) is the base, signifying the growth or decay factor (less than 1 for decay).
• \(t\) is the exponent, typically reflecting time or another progression metric.

In our specific example, \(y = 10(0.2)^t\), we see that this formula aligns perfectly with the understanding of exponential decay.
The presence of the base \(b\) dictates how the outcome changes over intervals, characterizing it either as decay or growth.

This form aids in foresight and planning, as it allows one to model and predict changes over a continuum, providing clear insights into dynamic processes.