The constant ratio of change, also known as the constant rate of change, describes a situation where the ratio between two quantities remains the same over the entire domain. In simpler terms, for every unit change in the input (often time), the output changes by a fixed amount.
Understanding Constant Rate of Change
The constant rate of change is most commonly associated with linear functions, where the graph is a straight line. The slope of this line represents the constant rate of change.
Formula
According to the provided reference, the formula for calculating the constant rate of change is:
(y2 − y1) / (x2 − x1)
Where:
- (x1, y1) and (x2, y2) are two points on the line.
- (y2 − y1) represents the change in the output (y-values).
- (x2 − x1) represents the change in the input (x-values).
- The ratio calculates how much the output changes for each unit change in the input. This is often related to time.
Examples
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Example 1: If a car travels at a constant speed of 60 miles per hour, the constant rate of change is 60 miles/hour. For every hour (input), the car covers 60 miles (output).
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Example 2: Consider a plant that grows at a constant rate. Let's say it grows 2 centimeters per week. The constant rate of change is 2 cm/week.
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Example 3: Finding the constant rate of change from two points. Say we have the points (1,5) and (3,11).
- Using the formula: (11-5)/(3-1) = 6/2 = 3. The constant rate of change is 3.
Practical Insights
- Linear Relationships: Constant rate of change is a key characteristic of linear relationships.
- Predictions: Knowing the constant rate of change allows us to make predictions about future values.
- Modeling Real-World Phenomena: Many real-world scenarios can be modeled using linear functions with a constant rate of change, such as simple interest calculations, constant speed motion, and uniform growth patterns.
Table illustrating Constant Rate of Change
| Input (x) | Output (y) | Change in x | Change in y | Ratio of Change (y/x) |
|---|---|---|---|---|
| 1 | 3 | - | - | - |
| 2 | 6 | 1 | 3 | 3 |
| 3 | 9 | 1 | 3 | 3 |
| 4 | 12 | 1 | 3 | 3 |
In this example, the ratio of change is consistently 3, indicating a constant rate of change.
