Question

Give two pieces of information that may be used to formulate an exponential growth or decay function.

Short answer

Answer: The two key pieces of information needed are the initial amount (a) and the growth or decay factor (b). The initial amount (a) is the starting value at time t = 0 and forms the base from which growth or decay occurs. The growth or decay factor (b) determines whether the function models growth (b > 1) or decay (0 < b < 1) and represents the rate at which the quantity increases or decreases over time.

Step-by-step solution

Initial Amount (a)

The initial amount, represented by the variable "a" in the exponential function f(t) = ab^t, is the starting value at time t = 0. It forms the base from which the growth or decay occurs. For example, if we are calculating the population of bacteria in a petri dish, the initial amount would be the number of bacteria present at the beginning of the observation.

Growth or Decay Factor (b)

The growth or decay factor, represented by the variable "b" in the exponential function f(t) = ab^t, determines whether the function models growth or decay. A growth factor occurs when b > 1, while a decay factor occurs when 0 < b < 1. The growth or decay factor represents the rate at which the quantity increases or decreases over time. For example, if we are analyzing the half-life of a radioactive substance, the decay factor would reflect the proportion of the substance that remains after each half-life period.

Key concepts

Understanding Initial Amount in Exponential Functions

When we talk about the initial amount in exponential functions, we're referring to the value that represents the starting point of the phenomenon being modeled. In the formula of an exponential function, typically written as \( f(t) = ab^t \), this initial amount is denoted by the variable \( a \). It is the value of the function when \( t = 0 \), that is, the moment we begin observing or measuring.

Let's consider a practical scenario. If we were to track the growth of a savings account with compound interest, the initial amount would be the sum of money deposited into the account before any interest is accrued. This 'starting pot' of money is crucial, as it ultimately influences how much the account will grow over time, assuming a constant rate of growth.

It is essential for students to grasp that regardless of how quickly or slowly the growth or decay occurs, the initial amount remains constant—it is the fixed starting line from which everything else progresses.

Deciphering the Growth or Decay Factor

The concept of a growth or decay factor is pivotal in exponential functions. It indicates whether an amount is multiplicatively increasing or decreasing over time. Represented by the variable \( b \) in the equation \( f(t) = ab^t \), this factor determines the behaviour of the function.

If \( b > 1 \), the function models exponential growth, which can be synonymous with the spreading of a virus, where each infected individual could spread the disease to more than one other person. Conversely, if \( 0 < b < 1 \), we witness exponential decay, similar to the depreciation of a new car's value as soon as it leaves the dealership.

An essential piece of understanding for students is to recognize that the growth or decay factor is multiplicative—it's not about adding or subtracting a fixed amount, but rather about repeatedly multiplying by a certain factor. For instance, a growth factor of 1.1 signifies a 10% increase in size for each step, while a decay factor of 0.5 means halving the value at every interval.

Modeling with Exponential Functions

The application of exponential functions is vast and spans across diverse fields such as biology, economics, and physics. Modeling with these functions allows us to predict and understand how quantities grow or shrink over time. By using the exponential function \( f(t) = ab^t \), we can model scenarios such as compound interest, population dynamics, radioactive decay, and more.

It is critical for students to learn how to construct these models by determining the initial amount and the growth or decay factor. For instance, predicting the future population of a species requires accurate knowledge of the current population and the rate of growth. With these parameters, an exponential model can simulate population changes over time. Additionally, by understanding the model, students can manipulate it to answer 'what if' questions, such as the impact of a change in growth rate on the population.

Rate of Increase or Decrease in Exponential Functions

The rate of increase or decrease is a crucial concept that describes how rapidly a quantity in an exponential function grows or decays. The rate can be derived from the growth or decay factor and is often expressed as a percentage. For instance, a growth factor of 1.05 suggests a 5% rate of increase per time period, while a decay factor of 0.85 indicates a 15% rate of decrease.

Understanding these rates helps to visualize the steepness of the exponential curve. In real-world scenarios, such as the interest rate of a savings account or the decline in the price of technology over time, these rates provide insight into how quickly we can expect to see significant changes. Students should learn how to calculate these rates from the growth or decay factor and to appreciate their impact on the exponential model's outcome over successive intervals.