8.3 Exponential Growth & Decay Models, & Comparing Function Families
Key Takeaways
- An exponential function has the algebraic form $f(x) = a \cdot b^x$, where $a$ represents the non-zero initial amount ($y$-intercept when $x = 0$) and $b$ represents the constant multiplicative base factor ($b > 0, b \neq 1$).
- In exponential growth models ($b > 1$), the growth factor is $b = 1 + r$, whereas in exponential decay models ($0 < b < 1$), the decay factor is $b = 1 - r$, where $r$ is the fractional growth/decay rate per time interval.
- Key real-world exponential formulas include compound interest $A = P\left(1 + \frac{r}{n}\right)^{nt}$, asset depreciation $V(t) = P(1 - r)^t$, and radioactive half-life decay $N(t) = N_0 \left(\frac{1}{2}\right)^{t / t_{1/2}}$.
- Function families are classified from data tables by inspecting rates of change: linear functions have constant first differences ($\Delta y$), quadratic functions have constant second differences ($\Delta^2 y$), and exponential functions have constant multiplicative ratios ($\frac{y_{n}}{y_{n-1}}$).
8.3 Exponential Growth & Decay Models, & Comparing Function Families
While linear functions model situations with a constant additive change, many real-world phenomena—including compound interest, biological population growth, epidemic spread, equipment depreciation, and radioactive decay—change by a constant multiplicative factor over equal intervals of time. These relationships are modeled by exponential functions.
1. Anatomy of an Exponential Function
The standard algebraic form of an exponential function is:
Anatomy of f(x) = a · b^x:
├── a: Initial Value / Starting Amount (y-intercept at (0, a) since b^0 = 1, a ≠ 0)
├── b: Base Factor / Multiplier (b > 0 and b ≠ 1)
└── x: Independent Variable (Exponent representing time intervals or cycles)
Key Structural Features
- Domain: All real numbers $(-\infty, \infty)$.
- Range (when $a > 0$): All positive real numbers $(0, \infty)$. The function values approach but never reach zero.
- Horizontal Asymptote: The line $y = 0$ ($x$-axis) acts as a horizontal asymptote.
2. Exponential Growth vs. Exponential Decay
The behavior of an exponential function is dictated entirely by its base factor $b$:
Exponential Growth vs Decay:
EXPONENTIAL GROWTH (b > 1) EXPONENTIAL DECAY (0 < b < 1)
| / | \
| / | \
| / | \
| / | \___
--------+---/---------- x --------+---------────── x
| / (Asymptote y=0) | (Asymptote y=0)
| Feature | Exponential Growth | Exponential Decay |
|---|---|---|
| Base Factor Condition | $b > 1$ | $0 < b < 1$ |
| Formulation with Rate $r$ | $b = 1 + r$ ($r = \text{growth rate as decimal}$) | $b = 1 - r$ ($r = \text{decay rate as decimal}$) |
| Curve Direction | Rises steeply from left to right | Falls rapidly, then levels off toward $y = 0$ |
| Real-World Examples | Compound interest, population growth | Machinery depreciation, drug clearance, half-life |
[!CAUTION] The Percentage Rate vs. Base Factor Trap: If an investment grows by $7%$ annually, the base factor is $b = 1 + 0.07 = 1.07$ (NOT $0.07$). If a vehicle depreciates by $15%$ annually, the base factor is $b = 1 - 0.15 = 0.85$ (NOT $0.15$ or $0.85$ subtracted from $x$).
3. Applied Exponential Growth & Decay Models
A. The Compound Interest Formula
When interest is compounded $n$ times per year, interest earned in earlier periods begins earning interest itself:
- $A = \text{Total accumulated balance (principal + interest)}$
- $P = \text{Principal (initial investment amount)}$
- $r = \text{Annual interest rate expressed as a decimal}$ ($6% = 0.06$)
- $n = \text{Number of compounding periods per year}$
- Annually: $n = 1$
- Semi-annually: $n = 2$
- Quarterly: $n = 4$
- Monthly: $n = 12$
- Daily: $n = 365$
- $t = \text{Time in years}$
Worked Example 1: A commercial reserve account deposits $$8,000$ at an annual interest rate of $6%$ compounded quarterly. Find the balance after $5$ years.
- Identify parameters: $P = 8000$, $r = 0.06$, $n = 4$, $t = 5$.
- Calculate periodic rate: $\frac{r}{n} = \frac{0.06}{4} = 0.015$.
- Calculate total periods: $nt = 4 \times 5 = 20$.
- Apply formula:
B. Equipment & Asset Depreciation
Physical machinery, commercial vehicles, and electronics lose value at an annual percentage rate:
Worked Example 2: A CNC manufacturing mill is purchased for $$45,000$ and depreciates at $12%$ per year. Find its value after $3$ years.
V(t) &= 45000(1 - 0.12)^t = 45000(0.88)^t \\ V(3) &= 45000(0.88)^3 = 45000(0.681472) = \$30,666.24 \end{aligned}$$ ### C. Radioactive Decay & Half-Life The **half-life** ($t_{1/2}$) is the time required for half of a radioactive substance to decay: $$N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}$$ - If a $200\text{ mg}$ sample has a half-life of $6\text{ hours}$, the amount remaining after $18\text{ hours}$ ($3$ half-life cycles) is: $$N(18) = 200\left(\frac{1}{2}\right)^{\frac{18}{6}} = 200\left(\frac{1}{2}\right)^3 = 200\left(\frac{1}{8}\right) = 25\text{ mg}$$ --- ## 4. Comparing Function Families: Linear, Quadratic, & Exponential On TABE Mathematics Level A, recognizing and comparing different function families from data tables and graphs is a heavily tested competency: ```text Tabular Difference Method for Function Identification (Equal x-intervals): ├── 1. First Differences Constant (Δy = k) ─────────> LINEAR (y = mx + b) ├── 2. Second Differences Constant (Δ²y = k) ───────> QUADRATIC (y = ax² + bx + c) └── 3. Ratios of Consecutive Outputs Constant ──────> EXPONENTIAL (y = a · b^x) ``` | Function Family | Standard Equation | Graphical Shape | Tabular Identification Rule | Long-Term Rate of Growth | | :--- | :--- | :--- | :--- | :--- | | **Linear** | $f(x) = mx + b$ | Straight line | **Constant 1st differences** ($\Delta y = c$) | Constant additive rate | | **Quadratic** | $f(x) = ax^2 + bx + c$ | U-shaped parabola | **Constant 2nd differences** ($\Delta^2 y = c$) | Accelerated polynomial growth | | **Exponential** | $f(x) = a \cdot b^x$ | J-shaped exponential curve | **Constant ratios** ($\frac{y_{k+1}}{y_k} = b$) | **Dominates all polynomial growth** | ### Step-by-Step Tabular Analysis Example Analyze the data table below where $x$ increases by uniform steps of $1$: | $x$ | $y$ | 1st Differences ($\Delta y$) | 2nd Differences ($\Delta^2 y$) | Output Ratio ($\frac{y_{k+1}}{y_k}$) | | :---: | :---: | :--- | :--- | :--- | | $0$ | $5$ | — | — | — | | $1$ | $11$ | $11 - 5 = \mathbf{6}$ | — | $11 / 5 = 2.20$ | | $2$ | $21$ | $21 - 11 = \mathbf{10}$ | $10 - 6 = \mathbf{4}$ | $21 / 11 \approx 1.91$ | | $3$ | $35$ | $35 - 21 = \mathbf{14}$ | $14 - 10 = \mathbf{4}$ | $35 / 21 \approx 1.67$ | | $4$ | $53$ | $53 - 35 = \mathbf{18}$ | $18 - 14 = \mathbf{4}$ | $53 / 35 \approx 1.51$ | - **First differences:** $6, 10, 14, 18$ (Not constant $\implies$ Not linear). - **Second differences:** $4, 4, 4$ (**Constant $+4 \implies$ Quadratic Function**). - **Ratios:** $2.20, 1.91, 1.67, 1.51$ (Not constant $\implies$ Not exponential). - **Conclusion:** The data are governed by a quadratic function (specifically $y = 2x^2 + 4x + 5$). --- ## 5. The Principle of Exponential Dominance A fundamental mathematical property tested on the TABE is that **an exponential growth function with base $b > 1$ will eventually exceed any linear or polynomial function**, regardless of how large the linear slope or polynomial coefficient may be. | $x$ | Linear: $L(x) = 500x$ | Quadratic: $Q(x) = 5x^2$ | Exponential: $E(x) = 2^x$ | | :---: | :---: | :---: | :---: | | $1$ | $500$ | $5$ | $2$ | | $5$ | $2,500$ | $125$ | $32$ | | $10$ | $5,000$ | $500$ | $1,024$ | | $15$ | $7,500$ | $1,125$ | $32,768$ (Exceeds Linear!) | | $20$ | $10,000$ | $2,000$ | $1,048,576$ (Exceeds Quadratic by 500x!) |A small business deposits $8,000 into a commercial reserve account earning an annual interest rate of 6% compounded quarterly. What will be the account balance after 5 years? (Round to the nearest cent.)
A data table records the following values: x: 0, 1, 2, 3, 4 y: 5, 11, 21, 35, 53 Which function family best models these data, and why?
A manufacturing company purchases a CNC milling machine for $45,000. The machine depreciates at a continuous annual rate of 12% per year. Which exponential equation models the machine's value V(t) after t years, and what is its estimated value after 3 years?