8.3 Exponential Growth & Decay Models, & Comparing Function Families

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.

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1. Anatomy of an Exponential Function

The standard algebraic form of an exponential function is:

$$f(x) = a \cdot b^x$$

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.

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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)

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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:

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

• $A = \text{Total accumulated balance (principal + interest)}$
• $P = \text{Principal (initial investment amount)}$
• $r = \text{Annual interest rate expressed as a decimal}$ ($6\% = 0.06$)
• $n = \text{Number of compounding periods per year}$
  • 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 $\text{\textdollar}8,000$ at an annual interest rate of $6\%$ compounded quarterly. Find the balance after $5$ years.

1. Identify parameters: $P = 8000$, $r = 0.06$, $n = 4$, $t = 5$.
2. Calculate periodic rate: $\frac{r}{n} = \frac{0.06}{4} = 0.015$.
3. Calculate total periods: $nt = 4 \times 5 = 20$.
4. Apply formula: $$A = 8000(1 + 0.015)^{20} = 8000(1.015)^{20}$$ $$A \approx 8000(1.346855) = \$10,774.84$$

B. Equipment & Asset Depreciation

Physical machinery, commercial vehicles, and electronics lose value at an annual percentage rate:

$$V(t) = P(1 - r)^t$$

Worked Example 2: A CNC manufacturing mill is purchased for $\text{\textdollar}45,000$ and depreciates at $12\%$ per year. Find its value after $3$ years.

$$\begin{aligned} V(t) &= 45000(1 - 0.12)^t = 45000(0.88)^t \\ V(3) &= 45000(0.88)^3 = 45000(0.681472) = \$30,666.24 \end{aligned}$$

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: $$N(18) = 200\left(\frac{1}{2}\right)^{\frac{18}{6}} = 200\left(\frac{1}{2}\right)^3 = 200\left(\frac{1}{8}\right) = 25\text{ mg}$$

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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:

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)

Step-by-Step Tabular Analysis Example

Analyze the data table below where $x$ increases by uniform steps of $1$:

• 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 $y = 2x^2 + 4x + 5$).

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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.