Family Overview

In Plain Language

Topic 2.2 connects sequences (2.1) to functions of real numbers:

• Linear $f(x)=b+mx$ is the continuous cousin of an arithmetic sequence $a_n=a_0+dn$ (add the same amount repeatedly). Point-slope $f(x)=y_i+m(x-x_i)$ matches $a_n=a_k+d(n-k)$.
• Exponential $f(x)=ab^x$ is the continuous cousin of a geometric sequence $g_n=g_0 r^n$ (multiply by the same factor repeatedly). Form $f(x)=y_i r^{x-x_i}$ matches $g_n=g_k r^{n-k}$.

Over equal-length input intervals: constant rate of change => linear; proportional change => exponential. Linear change is additive; exponential change is multiplicative. Sequences and their matching functions may use different domains (whole numbers vs reals). All four types can be pinned down from two distinct values.

Why It Matters

This is the bridge into Unit 2 exponential modeling: students must recognize which pattern they see and write the matching real-valued function, not only a sequence formula.

How Families Can Help

• Ask: "Are outputs adding a fixed amount, or multiplying by a fixed factor?"
• Practice: same two numbers as a sequence AND as $f(x)=b+mx$ or $ab^x$.
• Emphasize domain: dots on whole numbers vs a continuous curve for real $x$.

What Not to Do

• Do not treat sequence formulas and function formulas as identical without checking domain.
• Do not call proportional growth "linear."
• Do not forget that two points can determine these models.
• Do not mix additive slope language with multiplicative growth-factor language.