The common ratio in exponential functions is the constant factor that multiplies the previous output value to generate the next one. It's essentially the rate of change within the function.
Here’s a breakdown of this concept:
- Definition: According to our reference, the common ratio in an exponential function represents the rate of change. More specifically, it is the factor by which the function's output changes each time the input (usually 'x' or 't') increases by a single unit.
- How it works: If you have an exponential function in the form of f(x) = a * bx, then 'b' is the common ratio. When x goes up by 1, the previous value is multiplied by b.
Example of How the Common Ratio Affects Output:
| Input (x) | Output (f(x) = 2 * 3x) | Calculation | Output change relative to the previous |
|---|---|---|---|
| 0 | 2 | 2 * 30 = 2 * 1 | |
| 1 | 6 | 2 * 31 = 2 * 3 | * 3 |
| 2 | 18 | 2 * 32 = 2 * 9 | * 3 |
| 3 | 54 | 2 * 33 = 2 * 27 | * 3 |
In the above function, f(x) = 2 * 3x, the common ratio is 3. Notice that with each increase of 1 in x, the output is multiplied by 3.
- Growth vs. Decay:
- If the common ratio is greater than 1, the exponential function represents growth.
- If the common ratio is between 0 and 1, the function represents decay.
- If the common ratio is 1, the function is no longer exponential, but a horizontal line.
- Practical Application: In real-world scenarios, common ratios are crucial for understanding and predicting trends like compound interest, population growth, or radioactive decay.
Key Takeaway:
The common ratio is a fundamental characteristic of an exponential function. It determines how rapidly the output of the function changes relative to its input. It provides a clear understanding of the multiplicative growth or decay factor in these types of functions.
