No, a constant ratio is not linear; it is characteristic of exponential functions.
Let's break this down:
Linear vs. Exponential Functions
Understanding the difference between linear and exponential functions is key to answering this question.
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Linear Functions: These functions have a constant first difference. This means that for every equal interval in the input (x-value), the output (y-value) changes by a constant amount. The graph of a linear function is a straight line.
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Exponential Functions: Exponential functions, on the other hand, have a constant ratio. This means that for every equal interval in the input (x-value), the output (y-value) is multiplied by a constant factor. The graph of an exponential function curves.
Constant Ratio Explained
A constant ratio implies multiplicative growth or decay.
- Example: Consider the sequence 2, 4, 8, 16... Here, each term is multiplied by 2 to get the next term. The ratio between consecutive terms is consistently 2. This indicates exponential growth.
Constant Difference Explained
To further contrast, a constant difference signifies additive growth or decay, which is linear.
- Example: Consider the sequence 2, 4, 6, 8... Here, 2 is added to each term to get the next term. This indicates a linear relationship.
Summary Table
| Feature | Linear Function | Exponential Function |
|---|---|---|
| Growth Pattern | Constant Addition/Subtraction | Constant Multiplication/Division |
| Constant Property | Constant First Difference | Constant Ratio |
| Graphical Representation | Straight Line | Curve |
In conclusion, since linear functions are defined by constant first differences, not constant ratios, a constant ratio indicates a non-linear function, specifically an exponential function. The provided reference explicitly states that exponential functions have a constant ratio.
