Problem 36 - Algebra 2

Question

Exercises \(36-38\) , use the following information. Suppose you deposit a principal amount of \(P\) dollars in a bank account that pays compound interest. If the annual interest rate is \(r\) (expressed as a decimal) and the bank makes interest payments \(n\) times every year, the amount of money \(A\) you would have after \(t\) years is given by \(A(t)=P\left(1+\frac{r}{n}\right)^{n t}\). If the principal, interest rate, and number of interest payments are known, what type of function is \(A(t)=P\left(1+\frac{r}{n}\right)^{n t} ?\) Explain your reasoning.

Step-by-Step Solution

Answer

The function is an exponential function due to its form.

Step 1: Identify the Function Form

Look at the formula given for the amount of money after compounding interest over time: \(A(t)=P\left(1+\frac{r}{n}\right)^{nt}\). The expression resembles the form \(a^x\), where \(a\) is a constant base and \(x\) is a variable exponent. This form is typical of an exponential function.

Step 2: Confirm Identifying Features of an Exponential Function

In the formula \(A(t)=P\left(1+\frac{r}{n}\right)^{nt}\), the base of the exponent, \(\left(1+\frac{r}{n}\right)\), is a constant value when \(r\) and \(n\) are fixed. The variable exponent \(nt\) increases as time \(t\) progresses. This alignment with the structure \(f(x) = ab^x\) confirms it is an exponential function.

Step 3: Reasoning through Problem Statement

Given all parameters except \(t\) are constants (\(P, r, n\)), \(A(t)\) evolves with \(t\) according to the power of the exponent. An exponential function involves continuous multiplication by a fixed ratio \(\left(1+\frac{r}{n}\right)\) as time progresses, which mirrors the nature of compound interest accumulation.

Key Concepts

Compound Interest

Compound interest is an incredible concept in finance where you earn interest on both the initial amount of money you deposit and on the interest that accumulates over time. Unlike simple interest, which only earns on the original amount, compound interest allows your investment to grow faster because it generates interest on an ever-increasing total. Here's how it works:

• Principal Amount (P): The original amount you deposit or invest.
• Interest Rate (r): The percentage at which your money grows per period.
• Computation Frequency (n): How often the interest is calculated and added to your balance.

The formula to calculate the total amount with compound interest over time is:\[A(t) = P\left(1 + \frac{r}{n}\right)^{nt}\]This formula breaks down the growth into small steps, reinforcing the concept of exponential growth with each time period.

Interest Rate

The interest rate is a critical factor in any financial system, representing the cost of borrowing money or the payoff for investing. When it comes to compound interest, the interest rate dictates how quickly your balance grows over time.

• Expressed as a Decimal: In formulas and calculations, the interest rate is expressed as a decimal. For example, a 5% annual interest rate becomes 0.05.
• Annual vs. Periodic Rate: The annual rate is often divided by the number of compounding periods to find the periodic interest rate.

Understanding the interest rate allows you to predict how your investments will increase. A higher rate means faster growth because each period adds more interest. This makes the interest rate a powerful tool for financial planning and investment.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases with passing time, creating a curve that becomes steeper over time. It is the core concept behind compound interest, transforming linear growth into something much more powerful.

• Constant Base Rate: Each period multiplies your total by a constant factor, driving faster increases.
• Variable Exponent: Time acts as the exponent in our function, which significantly increases the total value over long periods.

In the compound interest formula, \( \left(1 + \frac{r}{n}\right)^{nt} \), the term \( \left(1 + \frac{r}{n}\right) \) acts as the base while \(nt\) is the exponent. As time \(t\) increases, the total grows rapidly, showcasing how exponential growth works in financial contexts like savings accounts and investment returns. Understanding this helps one recognize the profound impact exponential growth can have on wealth over time.