Construct and interpret exponential functions, including growth and decay models, and identify the initial value and the growth or decay factor from an equation or context (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard introduces the exponential function and its two headline applications: growth and decay. Unlike a linear function, which adds a constant, an exponential function multiplies by a constant base each step. The Georgia Milestones EOC asks you to build a model from a percent rate, read the initial value and the growth or decay factor from an equation, and evaluate and interpret the model. The growth and decay formulas are typically not on the reference sheet, so you must carry $y = a(1 + r)^t$ and $y = a(1 - r)^t$ in your head.

The exponential form

An exponential function is $y = ab^x$, where $a$ is the initial value (the output when $x = 0$, since $b^0 = 1$) and $b > 0$, $b \neq 1$, is the base or growth factor.

• If $b > 1$, the function grows (each step multiplies by more than 1).
• If $0 < b < 1$, the function decays (each step multiplies by less than 1).

The defining feature is a constant ratio: dividing any output by the previous one gives the same base $b$, in contrast to a linear function's constant difference.

Growth and decay from a percent rate

Most EOC items give a percent rate $r$ and ask for a model. Convert the percent to a decimal and build the base.

A 4 percent annual increase gives base $1 + 0.04 = 1.04$. A 15 percent annual loss gives base $1 - 0.15 = 0.85$.

Reading the initial value and rate

From an equation $y = a(1 \pm r)^t$:

• The initial value is $a$, the coefficient.
• The rate is the distance of the base from 1: base $1.04$ means $+4\%$; base $0.85$ means $-15\%$.

This reading is a frequent one-point item: "what is the growth rate" or "what does 500 represent."

How the Milestones examines this topic

• Multiple choice. Read the growth or decay rate, or the initial value, from a model.
• Numeric entry. Evaluate a model at a given time.
• Constructed response. Write a growth or decay model from a context and evaluate it, with the correct base.

Why the base is 1 plus or minus r, not r

The most damaging error in this topic is using the percent itself as the base, writing $24000(0.15)^t$ for a 15 percent loss. Thinking about one step explains why that is wrong. After one year the car has not lost everything and kept 15 percent; it has lost 15 percent and kept 85 percent. So the new value is $85\%$ of the old, that is multiplied by $0.85 = 1 - 0.15$. For growth, gaining 4 percent means the new amount is the whole old amount plus 4 percent more, that is $104\%$, or multiplied by $1.04 = 1 + 0.04$. The base is always the multiplier that takes you from one period to the next, which is $1 - r$ for a loss and $1 + r$ for a gain. Holding onto "what fraction remains (or results) after one step" makes the base obvious and prevents the $r$-as-base mistake.

Compound interest as exponential growth

Compound interest is a direct exponential model. An amount $P$ growing at annual rate $r$, compounded once a year, becomes $A = P(1 + r)^t$ after $t$ years, exactly the growth form with $a = P$. If interest compounds $n$ times per year, the model is $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where the rate per period is $\frac{r}{n}$ and the number of periods is $nt$. Recognizing compound interest as the growth formula in disguise means you do not need a separate procedure: identify the principal as the initial value and the per-period multiplier as the base, and the same exponential reasoning applies.

Try this

Q1. Write a model for a $2000 investment growing 6 percent per year, and find its value after 3 years. [2 points]

• Cue. $A(t) = 2000(1.06)^t$; A(3) = 2000(1.06)^3 \approx \2382.03$.

Q2. In $N(t) = 80(0.9)^t$, what is the decay rate? [1 point]

• Cue. Base $0.9 = 1 - 0.10$, so a 10 percent decay rate.