๐ Topic Summary
Exponential growth occurs when a quantity increases at a rate proportional to its current value. This means the larger the quantity, the faster it grows! It's often modeled by the equation $y = a(1 + r)^x$, where $y$ is the final amount, $a$ is the initial amount, $r$ is the growth rate (as a decimal), and $x$ is the number of time periods. Think of populations doubling, or investments growing rapidly โ that's exponential growth in action! It's different from linear growth, which increases by a constant amount each time.
๐งฎ Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Exponential Growth |
A. The initial amount before growth starts. |
| 2. Growth Rate |
B. A quantity increases by the same factor over equal intervals of time. |
| 3. Initial Value |
C. The percentage increase over a period of time, expressed as a decimal. |
| 4. Time Period |
D. The interval over which growth is measured. |
| 5. Exponential Decay |
E. A quantity decreases by the same factor over equal intervals of time. |
Matching Answers:
1-B, 2-C, 3-A, 4-D, 5-E
๐ Part B: Fill in the Blanks
Exponential growth is a phenomenon where the rate of increase becomes faster in relation to the growing total population or size. It's often modeled by the equation $y = a(1 + r)^x$, where '$a$' represents the __________, '$r$' is the __________ (as a decimal), and '$x$' is the number of __________. In contrast to exponential growth, __________ involves a decrease at a rate proportional to the current value.
Fill in the Blank Answers: initial value, growth rate, time periods, exponential decay.
๐ค Part C: Critical Thinking
Imagine you've discovered a new species of bacteria that doubles in population every hour. If you start with 10 bacteria, how many will you have after 5 hours? Explain how you arrived at your answer, and discuss any real-world limitations to this model of exponential growth.