Linear and exponential functions represent fundamentally different patterns of change. A linear function exhibits a constant rate of change, resulting in a straight-line graph, while an exponential function exhibits a rate of change proportional to its current value, resulting in a curved graph.
Here's a breakdown of the key differences:
Key Differences
| Feature | Linear Function | Exponential Function |
|---|---|---|
| Rate of Change | Constant (addition/subtraction) | Proportional to the current value (multiplication/division) |
| Graph | Straight Line | Curve |
| Equation Form | y = mx + b (where m is slope, b is y-intercept) | y = a * b^x (where a is initial value, b is growth/decay factor) |
| Behavior | Increases or decreases at a steady pace. The reference ([3:18]) mentions that "a linear graph has a nice straight line...always going this way". | Increases or decreases at an accelerating pace. |
Examples
- Linear: Imagine you earn \$10 per hour. Each hour you work, your total earnings increase by a constant \$10.
- Exponential: Imagine a population of bacteria that doubles every hour. The increase in population becomes larger with each passing hour because the growth is proportional to the current population size.
Characteristics of a Linear Function
- The graph is a straight line, either increasing or decreasing. ([3:18] "my line is always going this way you'll always continue to go this way if I'm going down he's always going this way.")
- It has a y-intercept where the line crosses the y-axis. ([3:18] "I have a nice y-intercept")
- The equation is often in the form y = mx + b.
