A constant ratio property, in the context of exponential relationships, describes the factor by which the y-value is multiplied when the x-value increases by 1. In simpler terms, it's the consistent multiplier applied to the output (y) for each unit increase in the input (x).
For example, consider an exponential function like y = 2x. According to the provided reference:
- When x = 1, y = 2.
- When x = 2, y = 4.
As x increases by 1 (from 1 to 2), the value of y is multiplied by 2 (from 2 to 4). Therefore, the constant ratio in this example is 2. For an increase of 1 in x, the number that y is multiplied by is the constant ratio.
Let's illustrate this with a table:
| x | y = 2x |
|---|---|
| 0 | 1 |
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
Notice how each time x increases by 1, y is multiplied by 2. This consistent multiplication factor is the constant ratio. The constant ratio is closely related to exponential growth or decay.
- Exponential Growth: If the constant ratio is greater than 1, the function represents exponential growth.
- Exponential Decay: If the constant ratio is between 0 and 1, the function represents exponential decay.
In essence, the constant ratio provides a concise way to characterize the behavior of exponential functions, indicating how quickly (or slowly) the output changes relative to the input.
