Exponential word problems deal with situations where a quantity increases or decreases by a constant percentage over a period of time. The general form of an exponential function is $y = a(1 + r)^t$, where $y$ is the final amount, $a$ is the initial amount, $r$ is the rate of growth or decay (expressed as a decimal), and $t$ is the time. Understanding how to identify these variables in a word problem is key to solving them.
When the rate $r$ is positive, it indicates exponential growth. When $r$ is negative, it indicates exponential decay. Real-world applications include population growth, compound interest, and radioactive decay. These problems often require careful reading to extract the correct values for $a$, $r$, and $t$, and then using the formula to find the unknown quantity.
Match each term with its definition:
| Term | Definition |
|---|---|
| 1. Exponential Growth | A. The initial amount before growth or decay. |
| 2. Exponential Decay | B. The time period over which growth or decay occurs. |
| 3. Initial Amount | C. A decrease in a quantity by a constant percentage over time. |
| 4. Rate of Change | D. An increase in a quantity by a constant percentage over time. |
| 5. Time Period | E. The percentage by which a quantity increases or decreases. |
(Answers: 1-D, 2-C, 3-A, 4-E, 5-B)
Complete the following paragraph using the words: exponential, rate, initial, time, decay.
An _________ function models situations where a quantity changes over _________. The _________ amount is the starting value, and the _________ determines how quickly the quantity grows or _________.
(Answers: exponential, time, initial, rate, decay)
Explain, in your own words, how the rate of growth or decay affects the final amount in an exponential word problem. Give an example.
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