Linear Growth and Linear Decay

Linear expressions model situations where a quantity increases (grows) or decreases (decays) by a constant amount per unit time.

Consider the linear function for the cost of a journey:

$C(d) = 100 + 60d$

where $C$ is the cost in rupees and $d$ is the distance in kilometres.

For every 1 km increase in $d$, the cost $C$ rises by exactly ₹60. This is the per-step rule of every linear polynomial — the leading coefficient is the per-step change. Because $a = +60 > 0$, this is linear growth.

A real-world image of this: a taxi meter ticking up steadily as the car covers ground.

Now consider a linear function for the height of water in a cylindrical tank:

$h(t) = 3 - 0.5t$

where $h$ is the height in metres and $t$ is the number of months since the start of summer.

For every 1-month increase in $t$, the height $h$ decreases by exactly 0.5 m. Because the leading coefficient is $a = -0.5 < 0$, this is linear decay.

A real-world image of this: water evaporating from an open tank in steady weekly amounts.

Same algebra, opposite story. The cost-of-journey was linear growth ($a = +60$). The water tank is linear decay ($a = -0.5$). Both follow $f(t) = at + b$ — the only difference is the sign of $a$.