Build and interpret exponential functions of the form f(x) = ab^x, including growth y = a(1+r)^t and decay y = a(1-r)^t, identifying the initial value and the rate (Ohio F-LE.1, F-LE.2, F-IF.8).

What this topic is asking

Ohio standards F-LE.2 and F-IF.8 ask you to build and interpret exponential functions. The base form is $f(x) = ab^x$, where $a$ is the initial value and $b$ is the growth factor. The percentage growth and decay models, $y = a(1 + r)^t$ and $y = a(1 - r)^t$, are not on the reference sheet, so you must memorize them. Exponentials sit in the Functions category and appear especially on the calculator part.

The base form

Every exponential function has the same two-part structure.

So $f(x) = 3 \cdot 2^x$ starts at $3$ and doubles each step; $g(x) = 80 \cdot (0.5)^x$ starts at $80$ and halves each step.

The percentage growth and decay models

When a quantity changes by a percent each period, convert the percent to the base.

Reading growth versus decay

The base alone tells the story. A base greater than $1$ means the quantity increases (growth); a base between $0$ and $1$ means it decreases (decay). Converting a percent: add for growth ($1 + r$), subtract for decay ($1 - r$). A base of $1.07$ is $7\%$ growth; a base of $0.92$ is $8\%$ decay (since $1 - 0.92 = 0.08$).

How Ohio examines this topic

• Equation response. Write the growth or decay function from a context, then evaluate it (often on the calculator part).
• Multiple choice and multiple-select. Pick the model whose base matches the percent, distinguishing growth from decay and from a linear model.
• Tables. Identify a constant ratio between outputs as the signature of exponential change.

Why exponential change is multiplicative

The defining feature of an exponential function is that it multiplies by a constant factor each step, while a linear function adds a constant amount each step. This is why a constant percent change is exponential: keeping or adding the same fraction of the current amount means multiplying by a fixed base, not adding a fixed number. A savings account earning $4\%$ grows by more dollars each year as the balance rises, because $4\%$ of a larger amount is larger, that increasing dollar growth is the hallmark of multiplication. Spotting "per year by a percent" (exponential) versus "per year by a fixed amount" (linear) is the key modeling decision, and the test builds distractors around exactly this difference.

Why growth eventually beats linear

Over a short interval a steep line can stay ahead of a slowly growing exponential, but in the long run exponential growth overtakes any linear function. Multiplying by a base greater than $1$ compounds: the increases themselves keep getting bigger, so the curve bends upward and accelerates, while a line rises at a fixed rate forever. This is why population, compound interest, and viral spread are modeled exponentially, and why "which is larger for large $t$?" almost always favors the exponential once $t$ is big enough. Understanding this contrast is the heart of comparing function families, and it explains why a constant-ratio table eventually dwarfs a constant-difference one.

Try this

Q1. Write a decay model for \1200$losing$10%$ per year. [2 points]

• Cue. Base $1 - 0.10 = 0.90$, so $V(t) = 1200(0.90)^t$.

Q2. For $f(x) = 5 \cdot 3^x$, what is $f(0)$? [1 point]

• Cue. $3^0 = 1$, so $f(0) = 5$.