The rate of change of a linear function is constant; it's the same everywhere on the line. This constant rate of change is also known as the slope of the line.
Explanation:
Linear functions are characterized by a straight-line graph. A crucial property of these functions is that their rate of change is constant. This means for any equal change in the input (x-value), there will always be an equal change in the output (y-value).
Key Concepts:
- Rate of Change: Describes how much a quantity (dependent variable) changes with respect to another quantity (independent variable).
- Constant Rate of Change: Indicates that the rate of change remains the same regardless of the points chosen on the function.
- Slope: The slope is a numerical value that represents the steepness and direction of a line. It is calculated as the change in y divided by the change in x (rise over run).
Formula:
The slope (m) is calculated as:
m = (y₂ - y₁) / (x₂ - x₁)
Where (x₁, y₁) and (x₂, y₂) are any two points on the line.
Example:
Consider the linear function y = 2x + 3.
- The rate of change (slope) is 2.
- For every increase of 1 in x, y increases by 2.
In summary, the rate of change for a linear function is constant, and is represented by its slope. This constant rate of change defines the consistent relationship between the input and output values of the function.
