Definition: A linear function is a function whose graph is a straight line, with a constant rate of change, written y = mx + b.
What is a linear function?
Its output changes by the same amount each time the input increases by 1. The graph is a straight line.
How do you recognize one?
In a table, equal steps in x give equal steps in y.
- x = 0, y = 4
- x = 2, y = 10
- x = 4, y = 16
Each 2-unit step adds 6, so the rate of change is 3 per unit and the function is y = 3x + 4.
How do you build one from a situation?
Mathbot’s tank holds 90 liters and drains at 6 liters per minute. Then V = 90 – 6t. The initial value is 90 and the rate of change is -6. It empties when 90 – 6t = 0, so at t = 15 minutes.
How is it different from a nonlinear function?
The function y = x2 gives 0, 1, 4, 9 for x = 0 to 3. The steps 1, 3, 5 are not constant, so it is not linear.
What is a common mistake?
Mixing up initial value and rate. The number multiplying x is the rate, and the constant added is the starting value.