Linear Growth and Linear Decay
When the leading coefficient is positive, you have growth. When it is negative, you have decay. Same algebra, opposite story.
Linear growth — Example 9 (NCERT cost-of-journey)
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:
where is the cost in rupees and is the distance in kilometres.
| (km) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| (₹) | 100 | 160 | 220 | 280 | 340 | 400 |
For every 1 km increase in , the cost rises by exactly ₹60. This is the per-step rule of every linear polynomial — the leading coefficient is the per-step change. Because , this is linear growth.
A real-world image of this: a taxi meter ticking up steadily as the car covers ground.
Linear decay — Example 10 (NCERT water tank)
Now consider a linear function for the height of water in a cylindrical tank:
where is the height in metres and is the number of months since the start of summer.
| (months) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| (m) | 3 | 2.5 | 2 | 1.5 | 1 |
For every 1-month increase in , the height decreases by exactly 0.5 m. Because the leading coefficient is , 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 (). The water tank is linear decay (). Both follow — the only difference is the sign of .
Formal definitions
Linear growth — Example 9 (NCERT cost-of-journey)
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:
where is the cost in rupees and is the distance in kilometres.
| (km) | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| (₹) | 100 | 160 | 220 | 280 | 340 | 400 |
For every 1 km increase in , the cost rises by exactly ₹60. This is the per-step rule of every linear polynomial — the leading coefficient is the per-step change. Because , this is linear growth.
A real-world image of this: a taxi meter ticking up steadily as the car covers ground.
Linear decay — Example 10 (NCERT water tank)
Now consider a linear function for the height of water in a cylindrical tank:
where is the height in metres and is the number of months since the start of summer.
| (months) | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| (m) | 3 | 2.5 | 2 | 1.5 | 1 |
For every 1-month increase in , the height decreases by exactly 0.5 m. Because the leading coefficient is , 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 (). The water tank is linear decay (). Both follow — the only difference is the sign of .