๐ Topic Summary
Exponential functions describe situations where a quantity increases or decreases at a rate proportional to its current value. They are written in the form $f(x) = ab^x$, where $a$ is the initial value, $b$ is the base (growth or decay factor), and $x$ is the exponent. If $b > 1$, the function represents exponential growth, and if $0 < b < 1$, it represents exponential decay. Understanding exponential functions is crucial in many fields, from finance and biology to computer science.
๐ง Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Exponential Growth |
A. The initial value of the function. |
| 2. Exponential Decay |
B. A function where the output decreases as the input increases. |
| 3. Base |
C. The value that is raised to the power of x in an exponential function. |
| 4. Initial Value |
D. A function where the output increases as the input increases. |
| 5. Exponent |
E. A quantity representing the power to which a given number or expression is to be raised. |
๐ Part B: Fill in the Blanks
Complete the following paragraph using the words: growth, decay, base, initial, exponential.
An ______ function is a function where the independent variable appears in the exponent. The general form of such a function is $f(x) = ab^x$, where 'a' is the ______ value, and 'b' is the ______. If the base 'b' is greater than 1, it represents _______, and if it is between 0 and 1, it represents _______.
๐ก Part C: Critical Thinking
Explain, in your own words, why exponential functions are important for modeling real-world phenomena. Give at least two examples.