3.09A - The Tangent Function - Basic, Transformed, Graphing

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High School
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High School Physics
Academic year:

2024
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Trigonometry Lessons 3.09A Tangent Functions – Basic, Transformed, Graphing Cycle Periodic 𝑓 𝑥 = tan 𝑥 𝑓 𝑥 = tan 𝑥 sin 𝑥 tan 𝑥 = cos 𝑥 = 0 for Zeros = 0 for Asymptotes 𝑓 𝑥 = tan 𝑥 • Period 𝑃=𝜋 𝑓 𝑥 = tan 𝑥 • 𝜋 3𝜋 5𝜋 , ,… 2 2 2 Asymptotes occur at … , , = 𝜋 2 + 𝜋𝑛 , 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℤ 𝑓 𝑥 = tan 𝑥 • It is important to note there is a “MIDPOINT OF THE CYCLE” halfway between the asymptotes Properties of Basic Function tan 𝑥 Domain ℝ\ 𝜋 2 + 𝜋𝑛 ; 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℤ Range ℝ Intercepts: 𝑓(0) 0 𝑓 𝑥 =0 Variation: 𝑥 = 0 + 𝜋𝑛 ; where 𝑛 ∈ ℤ Over the domain 𝑓↗ Extremes: Sign 𝐷𝑁𝐸 𝜋 𝑓≥0 0 + 𝜋𝑛, 2 + 𝜋𝑛 ; 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℤ 𝑓≤0 Vertex: Midpoint of Cycle Basic Ordered Pairs 𝜋 − 2 + 𝜋𝑛, 0 + 𝜋𝑛 ; 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℤ (0,0) 𝜋 𝜋 − , −1 , 0,0 , , 1 4 4 Properties of Transformed Function 𝑎 tan 𝑏 𝑥 − ℎ Domain ℝ\ Range ℎ 𝑃 + 2 +𝑘 + 𝑃𝑛 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℤ ℝ Intercepts: 𝑓(0) 𝑓 𝑥 =0 Variation: 𝑓 ↗ - strictly 𝑓 ↘ - strictly Extremes: Sign Vertex: Midpoint of Cycle Transformation of Basic Ordered Pairs Solve *REMEMBER* 𝑥 = 𝑥1 + 𝑃𝑛 ; where 𝑛 ∈ ℤ If 𝑎 and 𝑏 are same sign If 𝑎 and 𝑏 are opposite signs 𝐷𝑁𝐸 Solve between consecutive asymptotes (ℎ, 𝑘) 𝑥 + ℎ, 𝑎𝑦 + 𝑘 𝑏 Transforming Tangent 𝑓 𝑥 = 𝑎 tan 𝑏 𝑥 − ℎ +𝑘 • • • Summary: • 𝑏 affects the Period → 𝑃 = • • ℎ and 𝑘 affect the “MIDPOINT OF A CYCLE → ℎ, 𝑘 = “Vertex” If 𝑎 and 𝑏 are the same sign → strictly increasing If 𝑎 and 𝑏 are the opposite signs → strictly decreasing 𝜋 𝑏 (not 2𝜋 ) 𝑏 Asymptotes: 𝑃 ℎ+ + 𝑃𝑛, 𝑤ℎ𝑒𝑟𝑒 𝑛 ∈ ℤ 2 Graphing Tangent Functions - Steps 1. Find Period 𝑃 = 2. Find Asymptotes ℎ± 3. 𝜋 𝑏 𝑃 2 You should know right away whether: 𝑎, 𝑏 same sign Find 3 Points Using Basic Pairs and OPF: 𝜋 𝜋 − , −1 , 0,0 , , 1 4 4 4. Draw other Cycles 𝑥 𝑏 𝑎, 𝑏 opposite sign + ℎ, 𝑎𝑦 + 𝑘 Example: 𝑓 𝑥 = − tan 2 𝑥 − 𝑃= 2. Asymptotes 𝑃 2 𝜋 𝜋 𝜋 𝜋 2 ± = ± 4 2 4 4 𝑥1 = 0, 𝑥2 = 3. 𝜋 2 OPF 𝑥 + ℎ, 𝑎𝑦 + 𝑘 𝑏 𝜋 𝜋 − , −1 , 0,0 , , 1 4 4 4. +1 𝜋 2 𝜋 𝑏 1. ℎ± 𝜋 4 Draw other Cycles 𝑥 𝝅 + , −𝑦 + 𝟏 𝟐 𝟒 𝝅 𝝅 3𝝅 ,𝟐 , ,𝟏 , ,𝟎 𝟖 𝟒 𝟖 HW Visions 2 P107: 3be, 5