Study Guide

Learning Objectives

• Recognize sinusoidal forms $a\sin(b(\theta+c))+d$ and cosine analogs (3.6.A.1).
• Identify vertical shift $d$, phase shift $-c$, amplitude $|a|$, and period $2\pi/|b|$ (3.6.A.2–A.6).
• Graph or describe each transformation type separately and combined.
• Construct equivalent representations (equation $\leftrightarrow$ graph $\leftrightarrow$ verbal).

Suggested Practice Sequence

1. Vertical shift only: $y=\sin\theta+d$. 2. Amplitude only: $y=a\sin\theta$. 3. Period only: $y=\sin(b\theta)$; compute $2\pi/|b|$. 4. Phase shift only: $y=\sin(\theta+c)$; shift by $-c$. 5. Full package identification from an equation. 6. Match a graph to parameters; write an equation.

Classroom / Check-In Moves

• Parameter hunt: highlight $a,b,c,d$ in colored ink.
• "Rewrite $\sin(2\theta+\pi/3)$ into $b(\theta+c)$ form."
• Graph card match: four transforms, four graphs.

Misconception Watch-List

• Period $=2\pi b$ or $|b|/(2\pi)$.
• Phase shift sign errors.
• Amplitude signed as negative without interpreting reflection.
• Ignoring that cosine uses the same transform rules.

Vocabulary to Own

• amplitude; vertical shift; midline
• period; horizontal dilation
• phase shift; horizontal translation
• vertical dilation
• sinusoidal transformation form

Materials Map

• Practice: identify and apply $a,b,c,d$
• Word: describe transforms in plain language
• Mixed / multi-*: link to 3.5 characteristics
• Review & practice test: parameter fluency
• Interactive: form and period/phase checks