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. \(y=a\left(\frac{5}{4}\right)^{t / 22}\)

Short answer

The function rewritten in the form \(y = a(1 + r)^t\) is \(y = a \left(1+ \frac{1}{88}\right)^t\), and the growth rate is 1.1364% per time period.

Step-by-step solution

Rewrite the given exponential function

The function can be rewritten as \(y = a \left(1 + \frac{1}{4}\right)^{t/22}\) and multiplied through the exponent to become \(y = a \left(1+ \frac{1}{88}\right)^t\)

Identify the growth rate

The growth rate is the factor that the original value increases by each time period, represented by \(r\) in the function \(y = a(1 + r)^t\). In this case, \(r = \frac{1}{88}\), which is the growth rate per time period.

Format the growth rate into a percentage to ease understanding

Multiply the growth rate by 100 to represent it as a percentage: \(r = \frac{1}{88} * 100 = 1.1364\%\).

Key concepts

Growth Rate

Understanding the "growth rate" in an exponential function is key to interpreting how quickly something increases over time. Consider the equation \(y = a(1 + r)^t\), where:

• \(a\) is the initial value or starting amount,
• \(r\) is the growth rate, and
• \(t\) is the time period.

The growth rate \(r\) tells us by what percentage the initial amount grows in each time interval.
For example, if \(r = 0.02\), it means there's a 2% increase each period. A simple way to convert \(r\) into a percentage is by multiplying it by 100. If you have \(r = \frac{1}{88}\), as in the example, \(r = 0.011364\) or approximately 1.1364%.
This means that whatever you're measuring is increasing by about 1.1364% in every time period of \(t\). Recognizing this growth rate helps in predicting and understanding trends, whether in finance, population studies, or natural phenomena over time.

Exponential Growth

"Exponential growth" describes a pattern where the quantity increases at a rate proportional to its current value. This is in contrast to linear growth, where the amount added per period is constant.
The standard form for expressing exponential growth is \(y = a(1 + r)^t\), where the base of the exponent \((1 + r)\) determines the growth factor.

• If \(r > 0\), then \((1 + r) > 1\) and the function is growing.
• The larger the value of \(r\), the steeper the curve becomes, leading to a more rapid increase over time.

Exponential growth can model many real-world situations, such as:

• Population growth, where more individuals create more offspring.
• Compound interest, where investments grow based on accumulated interest.
• Biological processes, such as the spread of a virus.

Observing exponential growth in different realms helps us make informed predictions about future behavior of various systems.

Function Transformation

"Function transformation" involves altering a function's form to create a new version of the original equation.
In exponential functions, transformation often implies changing parameters to simplify or emphasize certain aspects of the equation. Take the function \(y = a\left(\frac{5}{4}\right)^{t/22}\).

• The transformation involves recognizing the base \(\frac{5}{4}\) as \(1 + \frac{1}{4}\).
• Rephrasing it as \(y = a(1 + \frac{1}{4})^{t/22}\) brings it closer to the standard growth form.
• Further, you can simplify \((1 + \frac{1}{4})^{t/22}\) into \((1 + \frac{1}{88})^t\), highlighting the growth rate per unit time.

By transforming functions, we make it easier to interpret and explore their characteristics. This technique allows us to uncover hidden properties or find more intuitive representations that facilitate analysis and application in real-world contexts.