3.08B - Intro to Transformed Sinusoidal Functions

Trigonometry Lesson 3.08B Transformed Sinusoidal Functions Cosine Sine Transforming the Rules 𝒇 𝒙 = 𝒂 𝐜𝐨𝐬 𝒃 𝒙 − 𝒉 +𝒌 𝒈 𝒙 = 𝒂 𝐬𝐢𝐧 𝒃 𝒙 − 𝒉 𝑎 ≠ 0, 𝑏 ≠ 0 "𝑎" • • Vertical Scale Change (See pg 294) VALUE – affects Amplitude 𝐴= 𝑎 • SIGN – reflection over 𝑥-axis 𝑉 𝑎+ 𝑉 𝑎− 𝑉 𝑉 +𝒌 Transforming the Rules 𝒇 𝒙 = 𝒂 𝐜𝐨𝐬 𝒃 𝒙 − 𝒉 +𝒌 𝒈 𝒙 = 𝒂 𝐬𝐢𝐧 𝒃 𝒙 − 𝒉 "𝑏" • • Horizontal Scale Change (See pg 295) VALUE – affects Period 𝑃= • 2𝜋 𝑏 or 𝑏 = 2𝜋 𝑃 SIGN – reflection over 𝑦-axis b+ b− 𝐶𝑂𝑆𝐼𝑁𝐸 is symmetrical over 𝑦-axis ∴ 𝑏 < 0 has no effect 𝑉 𝑉 +𝒌 Transforming the Rules 𝒇 𝒙 = 𝒂 𝐜𝐨𝐬 𝒃 𝒙 − 𝒉 +𝒌 𝒈 𝒙 = 𝒂 𝐬𝐢𝐧 𝒃 𝒙 − 𝒉 "ℎ" • • Horizontal Translation (See pg 296) Called a PHASE SHIFT "𝑘" • • Vertical Translation (See pg 296) This applies to the value of your SINUSOIDAL AXIS. 𝑦=𝑘 +𝒌 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Properties of Transformed 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 𝑎 sin 𝑏 𝑥 − ℎ Domain ℝ Range 𝑘 − 𝐴, 𝑘 + 𝐴 Intercepts: 𝑓(0) solve If 𝑘 > 𝑎 If 𝑘 ≤ 𝑎 𝑓 𝑥 =0 𝑥1 + 𝑃𝑛 Variation: AND 𝑥MIN + 𝑃𝑛, 𝑥MAX + 𝑃𝑛 𝑓↘ 𝑥MAX + 𝑃𝑛, 𝑥MIN + 𝑃𝑛 𝑘+𝐴 min 𝑘−𝐴 Vertex Amplitude (A) Period (P) 𝑥2 + 𝑃𝑛 𝑓↗ Extremes: max Sign 𝑛𝑜𝑛𝑒 solve Changes at the Zeros ℎ, 𝑘 ± 𝐴 ℎ, 𝑘 𝐴= 𝑎 2𝜋 𝑃= 𝑏 +𝑘 Notations • • Lower case: 𝑎, 𝑏, ℎ, 𝑘 will always be parameters 𝑛 is a variable used to represent the number of complete cycles ∴𝑛 ∈ℤ • • • P = Period A = Amplitude Standard notation for properties is to include _____ +𝑃𝑛 to account for the repetition in the cycles HW Visions 2 Page 107: # 1