The function of uniform growth y = mx + n.

If you substitute consecutively values of x into y = mx + n, say into y = 2x + 3, and change x by d= 0.2, you find:

or, more general,

If x increases always by the same amount d (0.2), then y also increases by the constant amount md (0.4). d can have any value, it can even be negative. The increase in y, that is md, is always proportional to d.

Conversely, if this is so, this behaviour must exist for every value of d :

y = mx + n is the function of uniform or linear growth.

Example: Bars made out of metal expand with their temperature. Their length is a function of the temperature. It is shown in Physics that the length of a bar at a given increase in temperature increases always by a corresponding constant amount. In other words, you have a uniform increase in length and the equation y=n+mx must apply. In the notation of Physics, the length of the bar is l, the temperature is t, and the required equation is l = mt + n. Two pairs of values (l,t) determine the constants. Let at t = 0, l = l₀, so that l ₀= m·0 + n, or n = l₀; for an increase in the temperature of 1º, let the length of the bar change by al₀, where a is called the coefficient of linear thermal expansion; then the length of the bar is al₀ + l₀. Substituting this value into l=mt+l ₀ yields l₀+a l₀=m·1+ l₀ or m = al₀, that is l = a l₀·t +l₀ or l = l₀(1+at).

The assumption of uniform growth cannot be valid for all temperatures! You only have to think of melting and evaporationof asubstance. The physical validity of the above formula is confined to the temperature range to which uniform growth in length applies.

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