Study Guide

Change in Tandem

This guide supports AP Precalculus topic 1.1: how inputs and outputs of a function vary together across representations, and how to identify increasing and decreasing behavior on an interval. This topic is assessed on the AP Exam.

What this standard means

Recognize that a function pairs each allowed input with exactly one output, and that inputs and outputs vary in tandem under the function rule
Express the same relationship graphically, numerically (tables), analytically (equations), and verbally
Decide whether a function is increasing on an interval: for all \(a < b\) in the interval, \(f(a) < f(b)\)
Decide whether a function is decreasing on an interval: for all \(a < b\) in the interval, \(f(a) > f(b)\)
Use ordered pairs or a graph to justify increasing/decreasing claims (not only “it looks like it goes up”)

Related study unit

Course unit hub: **Polynomial and Rational Functions** for longer worksheet tracks. Use this CED topic page for topic 1.1-specific practice, guides, and check-ins.

How to use the 20 practice sets

| Sets | When to use | | --- | --- | | 1–5 | Intro — match representations; read simple increasing/decreasing claims | | 6–10 | Core — tables, graphs, and verbal stories with clear intervals | | 11–15 | Mixed review — justify with inequalities; spot non-functions | | 16–20 | Stretch — denser multi-representation and interval analysis |

Pacing: 10–15 minutes per session. Always name the interval when you say increasing or decreasing.

How to practice (10–15 minutes)

1. Pick one relationship and write it as a sentence, a table of 4 pairs, and a quick sketch. 2. Choose two inputs \(a < b\) and compare \(f(a)\) and \(f(b)\). 3. State “increasing on …” or “decreasing on …” with the interval named. 4. Switch representations: given an equation, build a short table; given a table, write a verbal rule. 5. Ask: is this still a function? (one output per input)

Teacher quick-use (5 minutes)

1. Display a simple graph; students write increasing/decreasing on a labeled interval. 2. Give three ordered pairs; class decides if the data could come from an increasing function. 3. One verbal story: students say whether output rises with input. 4. Exit: each student writes one increasing inequality using \(f(a)\) and \(f(b)\).

Progress checklist

Mark when mostly true on two different days:

[ ] Matches the same function across at least two representations
[ ] Explains increasing with \(a < b \Rightarrow f(a) < f(b)\) on a named interval
[ ] Explains decreasing with \(a < b \Rightarrow f(a) > f(b)\) on a named interval
[ ] Rejects “increasing everywhere” when the graph only rises on part of the domain
[ ] Uses correct function language (input/output, domain, rule)

Common misconceptions

Calling a function increasing because it ends higher than it starts, even if it dipped in between
Confusing a relation that fails the vertical-line test with a function
Thinking “positive numbers in the table” means increasing
Mixing up which variable is input vs output in a story
Using slope language on intervals where average rate of change is not yet the focus (save that for 1.2+)

Quick checks

1. Table: \((1,3),\ (2,5),\ (3,7)\). Is an increasing model reasonable on \([1,3]\)? Why? 2. If \(a < b\) and \(f\) is decreasing on an interval containing both, which is larger: \(f(a)\) or \(f(b)\)? 3. Verbal: “Height of a ball after \(t\) seconds.” Is height always increasing in \(t\)? Explain.

Exit tickets

1. Write the definition of increasing on an interval using \(a\), \(b\), and \(f\). 2. Give one table that looks decreasing on \(\{0,1,2,3\}\). 3. Name two representations of the rule “output is three more than input.”

Challenge problems

1. A graph rises on \([0,2]\) and falls on \([2,5]\). Write true statements about increasing/decreasing. 2. Can a function be neither increasing nor decreasing on \([0,4]\)? Sketch or describe an example. 3. Translate: “For every pair of times, later time means lower temperature” into function language.

Enrichment questions

1. How is “increasing on an interval” different from “has a positive average rate of change on that interval”? 2. Why do AP stems insist on naming the interval? 3. When does a verbal description fail to define a function?

Math talk prompts

“What has to be true about every pair of inputs on this interval?”
“Which representation makes increasing/decreasing easiest to see? Why?”
“How would you convince a skeptic without saying ‘it looks like’?”

Vocabulary review

| Term | Student-friendly meaning | | --- | --- | | Function | Each allowed input has exactly one output | | Domain | Allowed input values | | Range | Output values the function actually takes | | Increasing on an interval | Larger inputs give larger outputs throughout that interval | | Decreasing on an interval | Larger inputs give smaller outputs throughout that interval | | Representation | Equation, table, graph, or verbal description of the same rule |

Review and practice tests

1. Start Review 1/10 after sets 1–3 feel comfortable. 2. Climb review tiers when justifications (not only answers) are clean. 3. Use Practice Test 4/10–6/10 for mid-topic checks. 4. Practice Test 10/10 is the mastery bar for 1.1.

Materials for this standard

Family Overview — plain-language support for families
Exam Strategy — MC/FRQ moves for Change in Tandem
Common Mistakes — traps specific to this topic
Practice Problems / Word / Mixed / Multi-* — 20 sets each
Review (10 tiers) and Practice Test (10 tiers)
Answer key and interactive packs