Linear Functions: Interpreting and Graphing a Straight Line

A linear function (affine function) is defined by f(x) = ax + b, where a and b are real constants. The domain is the set of all real numbers and so is the range. The coefficient a — called the slope or rate of change — indicates how much f(x) changes for each unit increase in x. If a = 3, every additional unit of x produces 3 more units of output. The coefficient b — called the y-intercept or initial value — is the value of f(0), the point where the function crosses the vertical axis.

Taxi fares are a classic example: if the base fare is $5.50 and each kilometre costs $2.20, the total cost is f(x) = 2.20x + 5.50. Here b = 5.50 (fixed charge) and a = 2.20 (cost per km). The conversion between Fahrenheit and Celsius is also a linear function: C = (F − 32) × 5/9, rewritten as C = (5/9)F − 160/9. Whenever a relationship has a fixed component and a component proportional to a quantity, you are looking at a linear function.

The idea of a constant rate of change is the core of a linear function. While quadratic and exponential functions have rates that vary as x changes, a linear function maintains the same pace across its entire domain. This property makes it ideal for modelling predictable, proportional situations.

Two points determine a line, so the minimum needed to draw the graph is to calculate f(x) for two different values of x. The easiest point is always x = 0: f(0) = b, giving the y-intercept directly. A natural second point is x = 1: f(1) = a + b. With these two points plotted in the coordinate plane, simply draw the line through them and extend it in both directions.

The sign of a determines the behaviour of the function: if a > 0 the function is increasing — the graph rises from left to right; if a < 0 it is decreasing — the graph falls; if a = 0 the function is constant and the graph is a horizontal line. The larger the absolute value of a, the steeper the line. Two graphs with the same a but different values of b are parallel lines, since they have the same slope but are shifted vertically.

The coefficient b shifts the line up (if b > 0) or down (if b < 0) without changing its slope. If you change f(x) = 2x + 1 to f(x) = 2x + 5, the slope stays the same but the entire line moves up by 4 units. This intuitive behaviour lets you quickly sketch variations of the same function without recalculating points.

The zero of a linear function is the value of x for which f(x) = 0 — where the line crosses the horizontal axis. Solving ax + b = 0 gives x = −b/a. This point has concrete interpretations depending on context: it may be the moment a debt is fully paid off, the point where two prices become equal, or the output level at which a business starts turning a profit.

Consider a company whose monthly fixed cost is $8,000 and each unit sold generates $40 of net revenue above variable costs. Monthly profit can be modelled as L(x) = 40x − 8,000, where x is the number of units sold. The zero occurs at x = 8,000/40 = 200: below 200 units the company runs a loss; above 200 it makes a profit. That break-even point is exactly the zero of the linear function.