📚 Topic Summary
Exponential functions describe situations where a quantity grows or decays by a constant percentage over equal periods of time. They are characterized by a variable exponent. Unlike linear functions, which have a constant rate of change (addition or subtraction), exponential functions involve a constant multiplicative factor. This means that for every fixed change in the input, the output is multiplied by a constant value.
This activity will help you distinguish exponential functions from other types of functions (like linear) by examining their tables, graphs, and equations. Understanding this difference is crucial for modeling real-world scenarios such as population growth, compound interest, and radioactive decay.
🧮 Part A: Vocabulary
Match the term to its definition:
| Term |
Definition |
| 1. Exponential Function |
A. The horizontal line that a graph approaches but does not cross. |
| 2. Growth Factor |
B. A function in the form $f(x) = ab^x$, where $a$ is the initial value and $b$ is the growth/decay factor. |
| 3. Decay Factor |
C. When $0 < b < 1$ in an exponential function, indicating a decreasing value. |
| 4. Asymptote |
D. When $b > 1$ in an exponential function, indicating an increasing value. |
| 5. Initial Value |
E. The starting value of a function, where $x = 0$. |
✍️ Part B: Fill in the Blanks
Complete the paragraph using the words provided: exponential, constant, growth, decay, initial.
An ______ function is one where the rate of change is not ______ but multiplicative. This can either represent _______ or _______. The _______ value represents the starting point of the function.
🤔 Part C: Critical Thinking
Explain, in your own words, how you can identify if a table of values represents an exponential function. What characteristics would you look for?