Question

Rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then state the growth or decay rate. \(y=a(4)^{t / 6}\)

Short answer

The rewritten function is \(y = a(1+3)^{6t}\). It represents a growth function with a growth rate of 300%.

Step-by-step solution

Understand the given function

The given function is \(y = a(4)^{t / 6}\). In this function, \(a\) is the starting value, 4 is the growth factor, \(t\) is the time, and \(t / 6\) is the number of growth periods.

Rewriting the function

The given function can be rewritten to the required format by expressing 4 as \(4 = 1 + r = 1+3\), and rewriting \(t / 6\) as a single variable \(t\). Therefore, we can rewrite the function as \(y = a(1+3)^{6t}\).

Identifying the growth or decay rate

Since here \(r=3\), and it is greater than 0. Hence, the equation is a growth equation with a growth rate of 3 or 300%.

Key concepts

Growth Rate

The growth rate is an essential concept in understanding how exponential functions behave over time. It describes the rate at which a quantity increases or decreases. In the context of exponential functions, the growth rate is represented by the variable \(r\) in the equations \(y=a(1+r)^t\) or \(y=a(1-r)^t\).

In these equations:

• \(a\) is the initial amount or starting value.
• \(1+r\) is the growth factor, where \(r\) is the growth rate.
• \(1-r\) would be used for a decay factor, where \(r\) represents the decay rate.
• \(t\) represents time or the number of periods over which growth or decay happens.

For a positive \(r\), the function experiences exponential growth, meaning it increases rapidly over time. For example, if \(r=3\), as in the given exercise, it equates to a growth rate of 3 or 300%, meaning the quantity triples in one time period.

Exponential Growth

Exponential growth represents a situation where growth accelerates rapidly over time, as opposed to linear growth where the increase is steady. This type of growth is best exemplified by populations or investments, where the amount grows proportionally to its current value.

In the exercise, the equation \(y = a(4)^{t / 6}\) is transformed to show the underlying exponential growth pattern. We observe a typical form \(y = a(1+r)^t\), revealing the mathematical nature of this function. Here, \(4\) serves as the growth factor, when rewritten as \(1+3\), indicating a substantial increase.

The unique aspect of exponential growth is that it is characterized by a constant percentage increase per unit time period. In our exercise, a growth rate of 3 confirms this rapid increase, meaning each cycle in time period \(t\) sees the value growing threefold, which highlights the defining feature of exponential growth.

Function Transformation

Function transformation is a process used to modify functions to reveal their property in a more understandable form. In the given exercise, the transformation of \(y = a(4)^{t / 6}\) into the format \(y = a(1+r)^t\) is a prime example of function transformation.

The process involves breaking down the original base, \(4\), to fit the \(1+r\) pattern which clarifies the exponential growth behavior. This helps identify the growth rate \(r\) as 3, paving the way to understand how the function scales over periods of time. Function transformations are useful because they can make complex expressions more intuitive.

By expressing \(t/6\) simply as \(t\), the transformation adjusts time to reflect how many times the growth factor is applied, crucially affecting the output of the function. This simplification enables a more straightforward visualization of how values change, and supports calculations related to growth predictions and data analysis.