Model and interpret exponential growth and decay using the form y = a(1 + r)^t for growth and y = a(1 - r)^t for decay, and evaluate to solve real-world problems (A.F.2).

What this topic is asking

This part of A.F.2 asks you to model exponential growth and decay with the percent-change form and to solve real-world problems by evaluating the model. On the Virginia Algebra I SOL these are Functions items: write the growth or decay function, compute a value after some time, or pick the right model. The growth and decay formulas are not on the Algebra I formula sheet, so memorize them. They appear as fill-in-the-blank and multiple choice.

The growth and decay models

Both are special cases of $f(x) = ab^x$, with the base written to show the percent change:

• Growth: $y = a(1 + r)^t$. The quantity increases by a fraction $r$ each period, so it becomes $(1 + r)$ times as large.
• Decay: $y = a(1 - r)^t$. The quantity decreases by a fraction $r$ each period, so it keeps $(1 - r)$ of itself.

Here $a$ is the initial amount, $r$ is the rate as a decimal, and $t$ is the number of periods (years, hours, etc).

Converting a percent rate to a multiplier

The single most important step is turning the percent into the base:

1. Write the percent as a decimal: $5\% = 0.05$, $15\% = 0.15$.
2. For growth, add to $1$: base $= 1 + r$. So $5\%$ growth gives a base of $1.05$.
3. For decay, subtract from $1$: base $= 1 - r$. So $15\%$ decay gives a base of $0.85$.

A base of $1.05$ means "increase by $5\%$" (you keep $100\%$ and add $5\%$); a base of $0.85$ means "decrease by $15\%$" (you keep $85\%$).

Reading the parts in context

SOL items reward interpreting the model:

• The coefficient $a$ is the starting value (at $t = 0$).
• A base greater than $1$ is growth; a base between $0$ and $1$ is decay.
• The rate $r$ is the percent change per period, recoverable from the base: a base of $1.08$ means $8\%$ growth, and $0.92$ means $8\%$ decay.

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

The reason the multiplier is $1 + r$ (not just $r$) is that each period you keep what you had and then change it. If a population grows $3\%$, next year it is the original $100\%$ plus another $3\%$, which is $103\%$ or $1.03$ times as large, so the base is $1 + 0.03$. Using $0.03$ as the base would say next year's population is only $3\%$ of this year's, a catastrophic collapse, not growth. The same logic runs in reverse for decay: losing $15\%$ leaves $85\%$, so you multiply by $0.85 = 1 - 0.15$, not by $0.15$. The "$1$" represents the whole amount you start each period with, and the "$\pm r$" is the change applied to it. This is also why these are exponential, not linear: the percent change applies to the current amount, which keeps shifting, so the actual change in units grows or shrinks over time, the hallmark of repeated multiplication.

How the SOL examines this topic

• Fill-in-the-blank. Write a growth or decay function and evaluate it after a given time.
• Multiple choice. Pick the correct model, with distractors using $r$ as the base or the wrong $\pm$.
• Table items. Match a context to a table of exponential values.

Try this

Q1. Write a growth model for \500$increasing$4%$ per year. [1 point]

• Cue. $y = 500(1.04)^t$.

Q2. A base of $0.90$ in a decay model means what percent decrease? [1 point]

• Cue. $1 - 0.90 = 0.10$, so $10\%$.