📚 Topic Summary
Exponential functions are a fundamental part of Algebra 2. They describe situations where a quantity increases or decreases at a constant percentage rate over a period of time. Understanding how to evaluate these functions is crucial for solving real-world problems related to growth and decay, such as population changes, compound interest, and radioactive decay.
Evaluating an exponential function involves substituting a given value for the variable (usually $x$) in the exponent and then simplifying the expression. The general form of an exponential function is $f(x) = ab^x$, where $a$ is the initial value, $b$ is the base (growth/decay factor), and $x$ is the exponent. Let's practice!
🧮 Part A: Vocabulary
Match the following terms with their definitions:
| Term |
Definition |
| 1. Exponential Function |
A. The value that is raised to a power. |
| 2. Base |
B. A function of the form $f(x) = ab^x$, where $a$ and $b$ are constants. |
| 3. Exponent |
C. The initial amount before growth or decay. |
| 4. Initial Value |
D. A rate at which a quantity increases exponentially |
| 5. Growth Factor |
E. The power to which a number is raised. |
Match the terms! (Example: 1-B, 2-A, etc.)
✍️ Part B: Fill in the Blanks
Complete the sentences below using the correct terms:
An exponential function has the general form $f(x) = ab^x$, where 'a' represents the __________ __________, and 'b' is the __________. The variable, $x$, is the __________. When b > 1, the function shows exponential __________, and when 0 < b < 1, it demonstrates exponential __________.
🤔 Part C: Critical Thinking
Explain, in your own words, how the value of 'b' in an exponential function $f(x) = ab^x$ determines whether the function represents exponential growth or exponential decay. Give a real-world example of each.