Linear functions are characterized by their constant rate of change and straight-line graphs. Here's a breakdown of their key features:
Understanding Linear Functions
A linear function is a function whose graph is a straight line. It can be represented in various forms, but the slope-intercept form is commonly used:
- Slope-Intercept Form: y = mx + b
- Where m is the slope and b is the y-intercept.
Key Features of Linear Functions
Here's a table summarizing the key features:
| Feature | Description | Example |
|---|---|---|
| Slope | The slope (m) represents the rate of change of the function. It indicates how much y changes for every unit change in x. A positive slope indicates an increasing function; a negative slope indicates a decreasing function; and a zero slope indicates a constant function. | y = 2x + 3 has a slope of 2 (positive). y = -x + 5 has a slope of -1 (negative). y = 4 has a slope of 0. |
| Y-Intercept | The y-intercept (b) is the point where the line crosses the y-axis (when x = 0). It represents the value of the function when the input is zero. | In y = 2x + 3, the y-intercept is 3. |
| X-Intercept | The x-intercept is the point where the line crosses the x-axis (when y = 0). It can be found by setting y = 0 and solving for x. | In y = 2x + 3, the x-intercept is -1.5 (found by solving 0 = 2x + 3). |
| Constant Rate of Change | Linear functions have a constant rate of change, meaning the slope is the same between any two points on the line. | The slope of y = 2x + 3 will always be 2, no matter which two points are chosen. |
| Straight Line Graph | The graph of a linear function is always a straight line. An increasing linear function results in a graph that slants upward from left to right and has a positive slope. A decreasing linear function results in a graph that slants downward from left to right and has a negative slope. A constant linear function results in a graph that is a horizontal line. | N/A |
Types of Linear Functions Based on Slope (Reference)
- Increasing Linear Function: Slants upward from left to right (positive slope).
- Decreasing Linear Function: Slants downward from left to right (negative slope).
- Constant Linear Function: Horizontal line (zero slope).
Examples
- y = 3x - 2 (Increasing linear function)
- y = -0.5x + 1 (Decreasing linear function)
- y = 5 (Constant linear function)
