5.2 Exponential Functions: Growth & Decay Models

Exponential Functions: Growth & Decay Models

Exponential functions model real-world phenomena where a quantity changes by a constant percentage or constant multiplicative factor over equal intervals of time. On the Digital SAT, exponential modeling questions test your ability to construct equations, interpret constants in real-world contexts, and convert between different time scales.

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1. Anatomy of the Exponential Model

The standard formula for an exponential function is written in one of two equivalent forms:

$f(t) = a \cdot b^t \quad \text{or} \quad f(t) = a(1 \pm r)^t$

+-----------------------------------------------------------------------------+
|                     ANATOMY OF AN EXPONENTIAL MODEL                         |
|                                                                             |
|                             f(t) = a · b^t                                  |
|                                    |   |                                    |
|             +----------------------+   +-----------------------+            |
|             v                                                  v            |
|     [ INITIAL VALUE (a) ]                             [ GROWTH FACTOR (b) ] |
|     - Value when t = 0                                - b = 1 + r (Growth)  |
|     - y-intercept on graph: (0, a)                    - b = 1 - r (Decay)   |
|     - Starting population / principal                 - b > 1: Increases    |
|                                                       - 0 < b < 1: Decreases|
+-----------------------------------------------------------------------------+

Parameter Breakdown:

1. Initial Value ($a$): The output when $t = 0$, represented by the point $(0, a)$ on the Cartesian plane. In context, $a$ represents the starting population, initial investment, original purchase price, or baseline quantity.
2. Base / Growth Factor ($b$): The multiplier applied for each single unit increase in $t$.
   • If $b > 1$, the function models Exponential Growth.
   • If $0 < b < 1$, the function models Exponential Decay.
   • $b$ can never be negative or zero in real-world exponential models.
3. Percent Rate of Change ($r$): The fractional or percentage increase/decrease per unit time:
   • Growth: $b = 1 + r \implies r = b - 1$
   • Decay: $b = 1 - r \implies r = 1 - b$
4. Independent Variable ($t$): The number of elapsed time periods (e.g., years, months, days, hours).

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2. Growth Factor vs. Decay Factor Conversion

The table below demonstrates how percentage changes translate into exponential base values ($b$):

[!IMPORTANT] Common Percent-to-Decimal Traps:

• A $5%$ increase is $b = 1 + 0.05 = 1.05$, NOT $1.5$ (which is a $50%$ increase).
• A $0.4%$ increase is $b = 1 + 0.004 = 1.004$, NOT $1.04$ (which is a $4%$ increase).
• A $30%$ decay means $70%$ remains ($b = 0.70$), NOT $b = 0.30$ (which would mean a $70%$ decay).

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3. Time-Scale Transformations & Exponent Rewriting

The Digital SAT frequently tests equivalent exponential equations where the time unit is modified. These questions rely on the Power of a Power property: $b^t = (b^{1/k})^{kt} = (b^k)^{t/k}$.

+-----------------------------------------------------------------------------+
|                    TIME-SCALE TRANSFORMATION METHODS                        |
|                                                                             |
|   ANNUAL TO MONTHLY CONVERSION:                                             |
|   f(t) = a · (1 + r)^t  ====>  f(t) = a · [ (1 + r)^(1/12) ]^(12t)          |
|                                          -----------------                  |
|                                            Monthly Base                     |
|                                                                             |
|   PERIODIC CHANGE (EVERY k YEARS):                                          |
|   f(t) = a · b^(t / k)                                                      |
|   - When t = k, exponent = 1 (one multiplication by b)                      |
|   - When t = 2k, exponent = 2 (two multiplications by b)                    |
+-----------------------------------------------------------------------------+

Case 1: Converting from Annual Rate to Monthly Rate

Suppose an investment grows by $12%$ per year: $f(t) = 1,000(1.12)^t$, where $t$ is years. To find the monthly growth factor:

1. There are $12$ months in a year, so elapsed time in months is $m = 12t$, which means $t = \frac{m}{12}$.
2. Substitute $t = \frac{m}{12}$: $f(m) = 1,000(1.12)^{m/12} = 1,000\left((1.12)^{1/12}\right)^m$.
3. Calculate the monthly base: $(1.12)^{1/12} \approx 1.009488$.
4. This reveals a monthly growth rate of approximately $0.95%$ per month.

Case 2: Periodic Change Every $k$ Units

If a population doubles every $5$ years: $P(t) = P_0 \cdot 2^{\frac{t}{5}}$

• When $t = 5$, exponent is $5/5 = 1 \implies P(5) = P_0(2)^1 = 2P_0$.
• When $t = 10$, exponent is $10/5 = 2 \implies P(10) = P_0(2)^2 = 4P_0$.
• When $t = 15$, exponent is $15/5 = 3 \implies P(15) = P_0(2)^3 = 8P_0$.

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4. Linear vs. Exponential: The Critical SAT Distinction

SAT questions frequently present tables or verbal scenarios and ask whether the data is best modeled by a linear or exponential function:

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5. Step-by-Step Worked SAT Exam Examples

Worked Example 1: Interpreting Exponential Parameters

Problem: The value of a high-performance computer server, in dollars, is modeled by $V(t) = 4,200(0.82)^{t/2}$, where $t$ is the number of years after purchase. Which statement best interprets the number $0.82$ in this context?

Step-by-Step Solution:

1. Identify the base: $b = 0.82$. Because $0.82 < 1$, this represents exponential decay.
2. Calculate the decay percentage: $r = 1 - 0.82 = 0.18 = 18%$.
3. Inspect the exponent: The exponent is $\frac{t}{2}$, which indicates that one full decay cycle occurs every time $t$ increases by $2$.
4. Conclusion: The value of the server decreases by $18%$ every $2$ years (or retains $82%$ of its value every $2$ years).

Worked Example 2: Constructing an Equivalent Monthly Model

Problem: A tech company's active user base $U(t)$, in thousands, is modeled by $U(t) = 150(1.24)^t$, where $t$ is the number of years since 2022. Which equivalent function reveals the approximate monthly growth rate as a constant base?

Step-by-Step Solution:

1. There are $12$ months in a year, so if $m$ is the number of elapsed months then $m = 12t$ and $t = \frac{m}{12}$.
2. Rewrite the expression using exponent properties: $U(t) = 150 \cdot (1.24)^t = 150 \cdot \left((1.24)^{1/12}\right)^{12t}$
3. Evaluate $(1.24)^{1/12}$ using a calculator: $(1.24)^{1/12} \approx 1.018087 \approx 1.0181$.
4. Substitute back into the equation: $U(t) \approx 150(1.0181)^{12t}$ This indicates an approximate monthly growth rate of $1.81%$ per month.

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6. Desmos Testing Strategies & Rapid Visual Validation

[!TIP] Testing Equivalent Exponential Forms in Desmos: When an SAT problem asks which rewritten exponential equation is equivalent to an original annual model (e.g., $f(x) = 150(1.24)^x$ vs. $g(x) = 150(1.0181)^{12x}$):

1. Graph Line 1: y = 150 * (1.24)^x
2. Graph Line 2: y = 150 * (1.0181)^(12x)
3. Click the gear icon to convert both to tables. Inspect values at $x = 0, 1, 2, 5$.
4. If the outputs match at all test values, the models are equivalent.

[!NOTE] Exponential Regression ($y_1 \sim a \cdot b^{x_1}$): If a problem provides a table of values $(x_1, y_1)$ and asks for the exponential equation that models them:

1. Click the + button in Desmos and insert a table. Enter the $(x_1, y_1)$ coordinate pairs.
2. On a new line, type y1 ~ a * b^(x1).
3. Desmos will compute the exact parameters $a$ and $b$ instantly.