Study Guide

Graphs of sequences

You are learning graphs of sequences in Pre-calculus · Honors (Introduction to calculus). Focus: Represent sequences graphically and connect graphs to recursive and explicit forms.

What this standard means

Rewrite between equivalent forms before solving
Check for extraneous solutions when required
Graph or describe key features of the parent relation
Explain one step of a solution in words
State domain restrictions that matter for this topic

How to use the 20 practice sets

| Sets | When to use | | --- | --- | | 1–5 | First week — explore together; short items on Graphs of sequences | | 6–10 | Core practice — main representations and fluent written work | | 11–15 | Mixed review — explain thinking; mix problem types for 7.2 | | 16–20 | Stretch — multi-step / word problems and mastery tasks |

Pacing: 10–15 minutes per session. Praise clear explanations, not speed alone.

How to practice (10–15 minutes)

1. Warm up with one short item on graphs of sequences. 2. Write a one-sentence plan before computing. 3. Show organized steps and box the final answer. 4. Check with substitution, a diagram, or an estimate. 5. Explain one sticky step in your own words.

Teacher quick-use (5 minutes)

1. Warm-up: class uses rewrite, solve, then check domain on one easy 7.2 item (board or oral). 2. Guided: name the method, try one item together, and have students point to each key step. 3. Independent: partners try a short graphs of sequences item from Practice sets 1–5. 4. Exit: each student whispers to a partner one sticky spot for 7.2 today.

Progress checklist for students

Mark when it is mostly true on two different days:

[ ] Rewrite between equivalent forms before solving
[ ] Check for extraneous solutions when required
[ ] Graph or describe key features of the parent relation
[ ] Explain one step of a solution in words

Map to practice bands: sets 1–5 → early checks; sets 6–10 → core graphs of sequences; sets 11–20 → mixed and stretch.

Common misconceptions

Address these early. Name the misconception, show a correct model, then have students retry a short item.

Domain errors (logs of nonpositives, division by zero) — See Common Mistakes for diagnosis and repair steps.
Extraneous roots from squaring or logging both sides — See Common Mistakes for diagnosis and repair steps.
Off-by-one in sequence indexing — See Common Mistakes for diagnosis and repair steps.
Mixing up inverse operations (exp vs log) — See Common Mistakes for diagnosis and repair steps.

Quick checks

Use as 2–5 minute formative checks mid-lesson. Students should finish independently when possible.

1. Write one clean example that shows you understand graphs of sequences. 2. True or false / spot-the-error: invent a common slip for 7.2 and fix it. 3. In one sentence, name the method you used (rewrite, solve, then check domain).

Exit tickets

End-of-lesson evidence of learning. Collect, sort into got-it / almost / reteach, and plan the next day.

1. Solve a short graphs of sequences item and box the answer with units if needed. 2. In one sentence: What rule or idea did you use for 7.2?

Challenge problems

Stretch tasks for students ready for more complexity after core practice.

1. Create a multi-step problem that uses graphs of sequences and one idea from Introduction to calculus. Solve it. 2. Find and fix an error: write a wrong solution for 7.2, then correct it with reasons.

Enrichment questions

Open-ended extensions that deepen understanding and connections across topics.

1. How does graphs of sequences help with other topics in Introduction to calculus? 2. Connect Graphs of sequences to Estimate limits from tables: what stays the same and what changes?

Math talk prompts

Use for turn-and-talk, whole-class discussion, or written reflection.

What is the ask in your own words?
How does rewrite, solve, then check domain help you on this topic?
What would happen if we skipped a required condition of 7.2?
How do you know your answer matches the question?
What do the key features mean in this context?

Vocabulary review

Revisit these terms with examples, non-examples, and student-friendly definitions.

domain restriction — Inputs that make an expression undefined or invalid.
extraneous solution — A candidate that fails when checked in the original.
inverse — An operation or function that undoes another.
equivalent form — A rewritten expression that represents the same relationship.

Review and practice tests

1. Start Review 1/10 when sets 1–3 feel comfortable. 2. Move up one level when a review is completed with little help. 3. Use Practice Test 4/10–6/10 for mid-topic check-ins. 4. Practice Test 10/10 is the mastery bar for 7.2.

[ ] Rewrite between equivalent forms before solving
[ ] Check for extraneous solutions when required
[ ] Graph or describe key features of the parent relation

See also Exam Strategy for readiness and test-day moves on this topic.

Materials for this standard

Parent Guide — home language, connections, try-together tasks
Exam Strategy — quiz / unit-test prep and traps
Common Mistakes — diagnoses and fixes for this topic
Practice Problems — 20 printable sets
Word Problems / Mixed Practice — additional practice bands
Review — 10 difficulty levels (1/10 easiest → 10/10 stretch)
Practice Test — 10 difficulty levels for progress checks
Answer key — step-by-step solutions
Interactive — quiz, mixed quiz, flashcards, typed practice sets