Linear functions: understanding straight lines

In short: A linear function has the form y = m · x + b, and its graph is a straight line. The gradient m tells you how much y increases when x goes up by 1, and b is the value of y at x = 0.

In more detail: Linear functions describe steady change, such as costs made up of a basic charge and a fixed price per unit. Any straight line can be determined uniquely from two points.

In detail

• Everyday example: A taxi ride costs a basic charge of €3 plus €2 per kilometre. Then y = 2x + 3; for 10 km you pay 2 · 10 + 3 = €23.
• Gradient: The gradient m shows how much y changes when x increases by 1. If m is positive the line rises; if m is negative it falls; with m = 0 it is horizontal.
• y-intercept: b is the point where the line crosses the y-axis, in the taxi example the basic charge of €3.
• Gradient from two points: For the points (1 | 5) and (3 | 9), m = (9 − 5) / (3 − 1) = 2. Substituting one point gives 5 = 2 · 1 + b, so b = 3.
• Root: The line crosses the x-axis where y = 0, i.e. at x = −b / m. For y = 2x + 3 that is x = −1.5.
• Proportional functions: If b = 0, the line passes through the origin and y is proportional to x. Doubling x then doubles y.
• Position of two lines: Lines with the same gradient are parallel. If two lines are perpendicular, the product of their gradients is −1.
• Point of intersection: You find where two lines cross by setting their equations equal and solving for x. Example: 2x + 3 = −x + 9 gives x = 2 and y = 7.
• Context: Linear functions build on solving equations and differ clearly from exponential growth (gleichungen-loesen-grundlagen, exponentielles-wachstum, diagramme-lesen-und-verstehen).