๐ Exponential Growth and Decay
SAT Math โ Complete Mastery Guide
1. Exponential vs. Linear โ The Critical SAT Distinction
๐ Linear Growth
- A constant AMOUNT is added or subtracted each period
- Formula: y = mx + b
- "Adds 50 each year" โ linear
- "Constant rate of 5 per hour" โ linear
- In tables: constant DIFFERENCE between consecutive values
๐ Exponential Growth/Decay
- A constant PERCENT (ratio) is multiplied each period
- Formula: A = Aโ ยท bแต
- "Increases 50% each year" โ exponential
- "Doubles every 3 hours" โ exponential
- In tables: constant RATIO between consecutive values
โ ๏ธ SAT TRAP โ Read Precisely!
- "increases BY 50" โ linear (adding a fixed amount)
- "increases BY 50%" โ exponential (multiplying by a ratio)
- "increases TO 150%" โ exponential
Linear vs. Exponential: Visual Comparison
Interactive visualization โ available in the full lesson
Recognizing from a Table
| t | Linear y | Difference |
|---|---|---|
| 0 | 100 | โ |
| 1 | 120 | +20 |
| 2 | 140 | +20 |
| 3 | 160 | +20 |
Constant difference โ Linear โ
| t | Exponential y | Ratio |
|---|---|---|
| 0 | 100 | โ |
| 1 | 120 | 1.2 |
| 2 | 144 | 1.2 |
| 3 | 172.8 | 1.2 |
Constant ratio โ Exponential โ
2. The General Exponential Formula
A = Aโ ยท bแต
A
Amount at time t
Amount at time t
Aโ
Initial amount (when t = 0)
Initial amount (when t = 0)
b
Growth/Decay factor
Growth/Decay factor
t
Time (number of periods)
Time (number of periods)
๐ Growth Form (b > 1)
A = Aโ(1 + r)แต
- r = growth rate as a decimal (r > 0)
- (1 + r) = growth factor
- 5% growth โ r = 0.05, factor = 1.05
- "Increases by 5%" means multiply by 1.05
๐ Decay Form (0 < b < 1)
A = Aโ(1 โ r)แต
- r = decay rate as a decimal (0 < r < 1)
- (1 โ r) = decay factor
- 15% decay โ r = 0.15, factor = 0.85
- "Decreases by 15%" means multiply by 0.85