Linear Function — Definition, Formula & Examples
A linear function is a function whose graph forms a straight line. Its output changes at a constant rate — for every equal increase in the input, the output increases (or decreases) by the same amount.
A linear function is a function of the form , where and are real constants. The constant represents the slope (rate of change), and represents the -intercept (the value of the function when ).
Key Formula
Where:
- = Output value (dependent variable)
- = Slope — the constant rate of change
- = Input value (independent variable)
- = y-intercept — the output when x = 0
How It Works
To work with a linear function, you need two pieces of information: the slope and the -intercept . The slope tells you how steep the line is — specifically, how much changes when increases by 1. A positive slope means the line rises from left to right; a negative slope means it falls. The -intercept tells you where the line crosses the -axis. Once you know and , you can plug any -value into to find the corresponding output.
Worked Example
Problem: A linear function has a slope of 3 and a y-intercept of −2. Write the function and find f(4).
Write the function: Substitute the slope and y-intercept into the formula.
Evaluate at x = 4: Replace x with 4 and simplify.
Answer: The function is , and .
Why It Matters
Linear functions model any situation with a constant rate of change, such as hourly wages, constant-speed travel, or unit pricing. They are the foundation for studying systems of equations in algebra and for understanding more complex functions like quadratics and exponentials.
Common Mistakes
Mistake: Confusing the slope with the y-intercept when reading the equation.
Correction: In , the number multiplied by is always the slope , and the standalone constant is always the -intercept . For example, in , the slope is 5, not 7.