Linear Function
Change the slope and the y-intercept, and steer the line
Summary
A linear function is the simplest kind of function, one whose graph is a straight line: f(x) = ax + b, or in equivalent notation y = mx + n. a (or m) sets the slope and b (or n) the y-intercept. Change the a and b sliders and watch in real time how the line changes direction.
In depth
A linear function is fully defined by two basic parameters: the slope and the y-intercept. The slope tells you how much y changes when x increases by one unit; geometrically it equals the tangent of the angle the line makes with the horizontal (m = tan θ). If a > 0 the line rises, if a < 0 it falls, and if a = 0 it is horizontal (a constant function). The y-intercept is the y value where the line crosses the y-axis, that is, f(0) = b. The slope between two points is calculated with m = (y₂ − y₁)/(x₂ − x₁). This simple structure is the most widely used mathematical model in science and engineering, from the relation between speed and time to Ohm's law, and from education-and-salary regressions to climate models.
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