Write a function that describes a relationship between two quantities, building a linear or exponential model from a context (LA A1: F-BF.A.1, F-LE.A.2).

What this topic is asking

Standards A1: F-BF.A.1 and F-LE.A.2 ask you to build a function that models a relationship, writing a linear or exponential rule from a description, a table, or a graph. On LEAP 2025 these are Type I and Type III items. The central decision is linear versus exponential: a constant added each step is linear, a constant multiplier each step is exponential.

Linear: constant amount added

A linear model adds the same amount each step. The starting value is the constant term; the per-step change is the slope.

Exponential: constant factor multiplied

An exponential model multiplies by the same factor each step. The starting value is $a$; the factor is $b$ (greater than $1$ for growth, between $0$ and $1$ for decay).

A growth of $r$ (as a decimal) gives factor $1 + r$; a decay of $r$ gives $1 - r$. These exponential model forms are not on the reference sheet.

Reading the type from a table

In a table with equally spaced inputs, look at the outputs: if consecutive outputs have a common difference (subtract to a constant), it is linear; if they have a common ratio (divide to a constant), it is exponential.

How LEAP examines this topic

• Equation response. Write the function from a described situation.
• Multiple choice. Pick the rule that matches a context, with linear-versus-exponential distractors.
• Type III modeling. Build the model, then use it to predict a value.

A clarifying idea: the word that decides the model is the kind of change. "Adds $12$ a day" (same amount) is linear; "grows 12 percent a year" (same factor) is exponential. Watch for it.

Why the kind of change determines the model

The choice between a linear and an exponential model comes down to what stays constant as the input increases, and recognizing it is the heart of F-LE.A.2. In a linear relationship the difference between successive outputs is constant: each step adds the same fixed amount, so the graph is a straight line and the rate of change never varies. In an exponential relationship the ratio between successive outputs is constant: each step multiplies by the same fixed factor, so the amount added grows (or shrinks) as the quantity itself grows. This is why savings under a fixed weekly deposit are linear, while a balance earning a fixed percent interest is exponential, the interest is a percent of a changing balance, so the dollar amount added increases every period. Reading a context for "same amount added" versus "same factor multiplied," or reading a table for a common difference versus a common ratio, tells you which model to build before you write a single symbol. Getting this distinction right is also what the comparing-function-families topic tests directly, because exponential growth eventually outpaces any linear growth precisely due to that compounding multiplier.

Try this

Q1. A savings account has $300 and the owner adds$25 a week. Write $f(w)$ for the balance after $w$ weeks. [2 points]

• Cue. Linear: $f(w) = 300 + 25w$.

Q2. A population of 500 grows 4 percent per year. Write $f(t)$. [2 points]

• Cue. Exponential: $f(t) = 500(1.04)^t$.