Exponential functions and growth/decay models
What this note covers
- The General Form y = abx — What Each Parameter Does
- Identifying and Distinguishing Growth from Decay
- Graphing Exponential Functions
- Worked Example 1 — Population Growth
- Worked Example 2 — Depreciation (Decay)
- Interpreting Exponential Graphs and Reading Values
- Connecting Exponential Models to Other Standard 2 Topics
7 sections · 12 key terms & formulas · 6 common mistakes
The General Form y = ab^x — What Each Parameter Does
Every exponential relationship studied in Mathematics Standard 2 is written in the form:
y = abx
Understanding the role of a and b separately is the foundation of this topic.
- a (the initial value / y-intercept): When x = 0, b0 = 1, so y = a × 1 = a. The value of a is always the y-intercept. In practical contexts it is the starting quantity — the population at time zero, the purchase price of a car, the original investment, and so on. a must be positive in all HSC Standard 2 contexts.
- b (the base / multiplying factor): Each time x increases by 1, y is multiplied by b. The base controls both the direction and the rate of change.
- If b > 1 → exponential growth (the multiplier is bigger than 1, so the quantity increases each period).
- If 0 < b < 1 → exponential decay (the multiplier is a fraction, so the quantity decreases each period).
- b = 1 gives a horizontal line (constant), not covered as exponential in this course.
- b ≤ 0 is not applicable in Standard 2 contexts.
- x (the independent variable): Usually represents time — years, months, days. It can also represent number of generations, number of doses, etc.
- y (the dependent variable): The quantity being modelled — population size, dollar value, bacterial count, etc.
The relationship between b and the percentage rate of change r is important for linking to financial contexts:
- Growth: b = 1 + r, where r is the growth rate as a decimal (e.g. 8% growth → b = 1.08).
- Decay: b = 1 − r, where r is the decay rate as a decimal (e.g. 15% depreciation → b = 0.85).
Identifying and Distinguishing Growth from Decay
In exam questions you will need to identify whether a given equation, table of values, or graph represents growth or decay. Use the following checklist:
| Feature | Exponential Growth (b > 1) | Exponential Decay (0 < b < 1) |
|---|---|---|
| Value of b | Greater than 1 (e.g. 1.08, 2, 1.5) | Between 0 and 1 (e.g. 0.85, 0.6, 0.5) |
| Graph direction | Rises steeply from left to right | Falls steeply, approaches x-axis from left to right |
| Successive ratio | Each y-value is larger than previous | Each y-value is smaller than previous |
| y-intercept | a (positive) | a (positive, same formula) |
| Horizontal asymptote | y = 0 (curve rises away from x-axis) | y = 0 (curve approaches but never touches x-axis) |
| Practical examples | Population, compound interest, bacterial growth | Depreciation, radioactive decay, cooling |
Quick identification from a table: Calculate the ratio of consecutive y-values (ynext ÷ ycurrent). If this ratio is constant and greater than 1, growth; if constant and less than 1, decay. A constant ratio (as opposed to a constant difference) is the hallmark of an exponential — not a linear — relationship.
Identifying from an equation: Rewrite in y = abx form if needed, then inspect b directly.
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