Write, evaluate, and interpret exponential functions that model growth and decay, identifying the initial value and the growth or decay factor and rate (MA.912.AR.5.4, MA.912.F.1.6).

What this topic is asking

MA.912.AR.5 asks you to model growth and decay with exponential functions, then interpret the parts. The models are $y = a(1 + r)^t$ for growth and $y = a(1 - r)^t$ for decay. These are not on the B.E.S.T. reference sheet, so you must memorize them, and the EOC tests writing, evaluating, and interpreting them in money, population, and depreciation contexts.

The two models

Here $a$ is the starting amount, $r$ is the rate written as a decimal (5 percent is $0.05$), and $t$ is the number of time periods. The quantity in parentheses is the factor that multiplies the amount each period.

The factor is the key

The base of the exponential is the growth or decay factor $b$:

• Growth: $b = 1 + r > 1$. For 8 percent growth, $b = 1.08$.
• Decay: $b = 1 - r$, between $0$ and $1$. For 20 percent decay, $b = 0.80$.

Reading the base tells you the behavior instantly: a base above $1$ grows, a base below $1$ shrinks. The factor $0.88$ means "keep 88 percent each year," which is the same as losing 12 percent.

How the B.E.S.T. EOC examines this topic

• Equation editor. Write a growth or decay function from a context, or compute a value at a given time.
• Multiple choice. Choose the correct model, with "rate as base" and growth-vs-decay distractors.
• Interpretation items. State the initial value, the rate, or the meaning of the factor.

A clarifying idea: exponential change is repeated multiplication, while linear change is repeated addition. Each period you multiply by the same factor, which is why a small percentage compounds into large change over many periods, the engine behind both compound interest and depreciation.

Why the factor, not the rate, is the base

A frequent error is putting the bare rate ($0.05$) in the base instead of the factor ($1.05$), and seeing why fixes it for good. After one growth period the new amount is the old amount plus $r$ times the old amount, that is $a + ra = a(1 + r)$. So the multiplier per period is $(1 + r)$, not $r$ alone; the "$1$" preserves the existing amount and the "$r$" adds the growth on top. For decay, you keep the old amount minus $r$ times it, $a - ra = a(1 - r)$, so the multiplier is $(1 - r)$. Using $0.05$ as the base would multiply the amount by $0.05$ each year, shrinking \2000 to \100 in one year, obviously wrong for 5 percent growth. Deriving the factor from "keep what you have, then adjust by $r$" guarantees you build the right base every time.

Connecting to geometric sequences

An exponential model is the continuous cousin of a geometric sequence: both multiply by a constant factor each step. The growth factor $b$ plays the role of the common ratio $r$ in $a_n = a_1 \cdot r^{n-1}$. The difference is mostly notation, sequences are indexed by whole-number terms, while exponential functions take any input $t$, but the multiplicative behavior is identical. Recognizing this link helps on the EOC, because a table that multiplies by a constant is exponential whether it is labeled a sequence or a function, and the same base-finding skill applies.

Try this

Q1. Write a model for $500 growing 4 percent per year. [1 point]

• Cue. $V = 500(1.04)^t$.

Q2. A sample of 80 mg of a substance decays 25 percent per hour. Write the model. [1 point]

• Cue. $A = 80(0.75)^t$ (factor $1 - 0.25 = 0.75$).