Family Overview

Pre-calculus · AP · Polynomial and Rational Functions

What this topic is

Change in Tandem means a function’s inputs and outputs move together under one clear rule. Each allowed input has exactly one output, and that pairing can be shown in four ways: a graph, a table (numerical), an equation (analytical), or a verbal description.

Students also learn to say when a function is increasing or decreasing on a named interval:

• Increasing on an interval: whenever the input gets larger, the output always gets larger. In symbols, if \(a < b\) on that interval, then \(f(a) < f(b)\).
• Decreasing on an interval: whenever the input gets larger, the output always gets smaller. If \(a < b\), then \(f(a) > f(b)\).

Try this at home (2 minutes): Look at the pairs \((1, 4)\), \((2, 7)\), \((3, 10)\). As the first number goes \(1 \to 2 \to 3\), the second goes \(4 \to 7 \to 10\). Ask: “Does this look increasing on \(\{1, 2, 3\}\)? Why?” Then invent one decreasing triple of pairs and check the same way.

This topic is assessed on the AP Exam.

Why it matters

• Everything later in Unit 1 (rates of change, polynomials, rationals) assumes students can already track how inputs and outputs move together across representations.
• AP items often give one form (table, graph, equation, or story) and ask for another—or ask whether the function is increasing or decreasing on a stated interval.
• Naming the interval early prevents the common habit of saying “it’s increasing” when a graph only rises on part of the domain.
• Clear tandem language keeps “goes up,” slope, average rate of change, and end behavior from getting mixed together too soon.
• Families who ask “when \(x\) gets bigger, what happens to \(y\)?” are already practicing the core idea of this topic.

How you can support

• Ask: “If the input increases, does the output go up, go down, or sometimes both—and on which interval?”
• Have them show the same relationship two ways (table and sentence, or equation and quick sketch).
• Point to two points with \(a < b\) and ask which must be true: \(f(a) < f(b)\) or \(f(a) > f(b)\).
• Praise a correct “always up / always down on this interval” explanation even if arithmetic still needs fixing.
• Keep sessions short: one table plus one justification, or one story plus one equation.
• If they use a calculator graph, still ask for two ordered pairs that justify increasing or decreasing.

What not to do

• Do not jump to end behavior, holes, or asymptotes here—those are later Unit 1 topics.
• Do not treat “goes up then down” as increasing on the whole domain; ask for the interval.
• Do not let them memorize “positive slope = increasing” without checking an actual interval (especially on a nonlinear graph).
• Do not solve every matching-representations item for them; leave the justification step.
• Do not skip asking whether a verbal story is even a function (one output per input).