Family Overview

In Plain Language

Topic 1.2 defined average rate of change (AROC). Topic 1.3 asks how those rates behave for linear and quadratic functions.

• For a linear function $f(x)=mx+b$, the AROC on any interval $[a,b]$ equals the constant slope $m$.
• For a quadratic $f(x)=Ax^2+Bx+C$, AROCs over consecutive equal-length input intervals form a linear pattern: they change by a constant amount from interval to interval.
• AROC is still the slope of the secant through $(a,f(a))$ and $(b,f(b))$.

Why It Matters

AP Precalculus uses these patterns to tell function types from tables and to describe how rates themselves are changing. Later polynomial work builds on "rates of rates." Students who only compute one AROC and stop miss the CED focus on constancy (linear) vs constant change of AROC (quadratic).

How Families Can Help

• Ask: "If I change the interval length, does the linear AROC stay the same?"
• For a quadratic table with equal $x$-steps, compute several consecutive AROCs and look for an arithmetic pattern.
• Ask: "Are the rates changing? If so, by how much each step?"

What Not to Do

• Do not assume every function has constant AROC (only linear ones do).
• Do not mix unequal interval lengths when claiming the quadratic AROC pattern.
• Do not confuse "AROC is constant" with "the function values are constant."
• Do not treat change-in-AROC as a derivative unless the course has introduced that language carefully.