Write exponential functions of the form $f(x) = ab^x$ to model growth and decay, interpret the meaning of $a$ and $b$ in context, and determine whether a situation represents exponential growth or decay (TEKS A.9B, A.9C, A.9D).

What this topic is asking

Exponential functions are about 10 percent of the STAAR Algebra I test, the smallest function category but a clear discriminator above Approaches. TEKS A.9B, A.9C, and A.9D ask you to write an exponential function $f(x) = ab^x$, interpret the parameters $a$ and $b$ in context, and decide whether a situation is growth or decay. The formula is not on the reference sheet, so it must be memorized.

The form f(x) = ab^x

Every Algebra I exponential model fits $f(x) = ab^x$. The two parameters carry the meaning:

• $a$, the initial value. It is $f(0) = a \cdot b^0 = a$, the starting amount before any change.
• $b$, the growth or decay factor. It is the number each value is multiplied by for each unit increase in $x$.

For $f(x) = 300(2)^x$, the initial value is 300 and the quantity doubles ($b = 2$) each step.

Growth versus decay: the value of b

The base $b$ decides whether the quantity grows or shrinks, and its size encodes the percent change.

So $b = 1.08$ is 8% growth, and $b = 0.92$ is 8% decay. A base greater than 1 grows; a base between 0 and 1 decays.

Writing a model from a description

For decay, swap to $b = 1 - r$: a substance losing 10% per day from 50 grams is $A(t) = 50(0.90)^t$.

How STAAR examines this topic

• Multiple choice. Interpret $a$ or $b$, or classify growth versus decay from the base. The "rate versus factor" and "growth versus decay" confusions are standard distractors.
• Equation editor. Write the function from a description; the decay factor $1 - r$ is the most-tested piece.
• Inline choice. Choose growth or decay, and identify the initial value or factor from dropdowns.

A clarifying idea is that exponential change is multiplicative: each step multiplies by $b$, unlike linear change which adds a fixed amount. That is why a percent change becomes a factor $1 \pm r$ rather than a slope, and why the quantity curves rather than following a straight line.

Reading the factor back as a percent

STAAR often runs this in reverse, giving a function and asking for the percent change. From the base $b$, the rate is $r = b - 1$ as a decimal, then converted to a percent. A model $f(x) = 800(1.07)^x$ has $b = 1.07$, so $r = 0.07$, a 7% growth rate. A decay model $f(x) = 800(0.91)^x$ has $b = 0.91$, so $r = 1 - 0.91 = 0.09$, a 9% decay rate. The trap is reading $0.91$ as "91% decay"; the base is what remains after the loss, so the rate is $1 - b$ for decay, and $b - 1$ for growth. Practicing this conversion both ways, description to function and function to description, covers the full range of how the standard is assessed.

Doubling and halving

Two special bases are worth recognizing on sight. A quantity that doubles each period has $b = 2$, as in $f(x) = a(2)^x$, and one that triples has $b = 3$. A quantity that halves each period (a half-life) has $b = \frac{1}{2} = 0.5$, as in $f(x) = a(0.5)^x$. These describe growth and decay without an explicit percent, and translating "doubles" to $b = 2$ or "halves" to $b = 0.5$ is a quick win on items that use that wording instead of a rate.

Try this

Q1. Write a model for $1,200 growing 6% per year. [1 point]

• Cue. $f(t) = 1200(1.06)^t$.

Q2. A 40 mg dose decays 25% per hour. Write the model and state growth or decay. [1 point]

• Cue. $A(t) = 40(0.75)^t$; decay ($b < 1$).