📚 Topic Summary
Exponential functions model situations where a quantity increases or decreases at a constant percentage rate over time. These functions are often used to describe population growth, radioactive decay, and compound interest. The general form of an exponential function is $y = ab^x$, where $a$ is the initial value, $b$ is the growth/decay factor, and $x$ is the independent variable (usually time).
When solving word problems involving exponential functions, it's crucial to identify the initial value, the growth/decay factor, and the time period. Pay close attention to the units and make sure they are consistent throughout the problem. Practice identifying the key information in the problem statement, setting up the exponential function, and solving for the unknown variable.
🧠 Part A: Vocabulary
Match the terms with their definitions:
| Terms |
Definitions |
| 1. Exponential Growth |
A. The initial amount of a quantity before growth or decay starts. |
| 2. Exponential Decay |
B. A function where the output decreases by a constant percentage each time the input increases by one. |
| 3. Initial Value |
C. The constant percentage change in the function. |
| 4. Growth/Decay Factor |
D. A function where the output increases by a constant percentage each time the input increases by one. |
| 5. Rate of Change |
E. The factor by which a quantity changes over time; greater than 1 for growth, less than 1 for decay. |
📝 Part B: Fill in the Blanks
Exponential functions are used to model situations with consistent percentage changes. The general form is $y = ab^x$, where 'a' represents the _______, 'b' represents the _______, and 'x' is often _______. If 'b' is greater than 1, we have exponential _______. If 'b' is less than 1, we have exponential _______.
💡 Part C: Critical Thinking
Explain in your own words how the growth/decay factor 'b' in an exponential function affects the long-term behavior of the function. Provide examples to illustrate your explanation.