Study Guide

Learning Objectives

• Construct tan from the unit circle as slope (3.8.A.1).
• Relate tan to sin/cos (3.8.A.2).
• Describe period, asymptotes, and shape (3.8.B.1-B.3).
• Apply additive and multiplicative transformations (3.8.C.1-C.5).

Suggested Practice Sequence

1. Unit circle: read tan as slope; compute sin/cos ratio. 2. Graph y = tan theta: asymptotes, period pi, increasing branches. 3. Single transforms: +d, +(theta + c), a tan, tan(b theta). 4. Combined form: read |a|, pi/|b|, d, -c. 5. Context/slope problems and technology checks.

Classroom / Check-In Moves

• "Slope, ratio, asymptote" three-word check.
• One-branch sketch with two asymptotes labeled.
• Phase sign exit ticket: tan(theta + c) shifts -c.

Misconception Watch-List

• Period 2 pi instead of pi.
• Asymptotes at k pi instead of pi/2 + k pi.
• Phase shift sign (+c vs -c).
• Using tan when cos = 0.

Vocabulary to Own

• terminal ray; slope; asymptote
• period; phase shift; vertical dilation
• undefined; pi/2 + k pi
• y = a tan(b(theta + c)) + d

Materials Map

• Practice: definitions, traits, transforms
• Word: slope/angle contexts
• Mixed / multi-*: link to 3.2-3.6
• Review & practice test: asymptote/period fluency
• Interactive: trait and transform checks