Build or use an exponential model; interpret initial value and growth/decay factor in context. | Nonlinear functions | SAT Math

Working with exponential equations and understanding what each part means in a real scenario.

核心知识

In $𝑓 (𝑥) = 𝑎 \cdot 𝑏 𝑥$ , the value $𝑎$ is the initial amount (when $𝑥 = 0$ ) and $𝑏$ is the growth/decay factor per unit of $𝑥$ . Growth means $𝑏 > 1$ ; decay means $0 < 𝑏 < 1$ . A percent change of r% corresponds to $𝑏 = 1 + 𝑟 100$ (growth) or $𝑏 = 1 - 𝑟 100$ (decay).

深入理解

Exponential models show up whenever a quantity changes by a fixed percentage each period — population, investments, depreciation, half-life.

The SAT tests two main skills here:

Building the model. If a car worth $20,000 loses 15% of its value each year, the model is $𝑉 (𝑡) = 20, 000 \cdot (0.85) 𝑡$ . The 0.85 comes from $1 - 0.15$ .

Interpreting parts. Given $𝑃 (𝑡) = 500 \cdot (1.03) 𝑡$ , the 500 is the starting population and 1.03 means a 3% increase per time unit. If the equation is written as $𝑃 (𝑡) = 500 \cdot (1.03) 2 𝑡$ , the effective per-unit-of- $𝑡$ factor is $(1.03) 2 = 1.0609$ , which is about 6.09% growth.

Watch the exponent carefully. If the model uses $𝑏 𝑥 𝑘$ , the factor $𝑏$ applies every $𝑘$ units, not every single unit.

分步讲解

Identify the initial value (the amount when the independent variable is 0) → this is $𝑎$ .
Determine the percent change per period and convert to a multiplier: growth of r% → $𝑏 = 1 + 𝑟 100$ ; decay of r% → $𝑏 = 1 - 𝑟 100$ .
Check the exponent for any scaling (e.g., $𝑡 2$ means the factor applies every 2 units).
Write the model as $𝑓 (𝑥) = 𝑎 \cdot 𝑏 𝑥$ (adjusting the exponent if needed).

常见误解

Writing decay as a negative base (e.g., $(- 0.85) 𝑡$ ) instead of a base between 0 and 1, such as $0.85 𝑡$ .
Confusing the growth factor with the growth rate: a factor of 1.05 means a rate of 5%, not 105%.
Ignoring exponent scaling - $(1.06) 𝑡 3$ is not the same rate as $(1.06) 𝑡$ .

题目

示例解析

The value of a certain painting, in dollars, $𝑡$ years after it was purchased is modeled by $𝑉 (𝑡) = 8, 000 \cdot (1.12) 𝑡$ . What does the number 1.12 represent in this context?

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