3.08D - Finding Rule of Sinusoidal | Lecture Note

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2024
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Chapter 3: Trigonometry Lessons 3.08D Finding Rule of Sinusoidal Functions Vertex Period SINE 𝑓 𝑥 = 𝑎 sin 𝑏 𝑥 − ℎ • +𝑘 𝑎𝑏 > 0 2 1. ℎ, 𝑘 2. ℎ + ,𝑘 + 𝐴 3. ℎ+ 𝑃 ,𝑘 2 4. ℎ+ 3𝑃 ,𝑘 4 5. ℎ + 𝑃, 𝑘 𝑃 4 1 3 5 4 −𝐴 COSINE 𝑓 𝑥 = 𝑎 cos 𝑏 𝑥 − ℎ +𝑘 • 𝑎>0 1 1. 5 ℎ, 𝑘 + 𝐴 𝑃 ,𝑘 4 𝑃 ,𝑘 − 𝐴 2 3𝑃 ,𝑘 4 2. ℎ+ 3. ℎ+ 4. ℎ+ 5. ℎ + 𝑃, 𝑘 + 𝐴 4 2 3 Helpful Properties • Range 𝑘 − 𝐴, 𝑘 + 𝐴 [𝑚𝑖𝑛, 𝑚𝑎𝑥] • A = (y_max – y_min)/2 • k = (y_max + y_min)/2 • Sinusoidal Axis 𝑦=𝑘 • Amplitude: 𝐴= 𝑎 • Period • Vertex Sine: ℎ, 𝑘 Cosine: ℎ, 𝑘 + 𝑎 2𝜋 𝑃= 𝑏 Finding rule: steps • Solve A and k from y_min and y_max • Solve b from period • Choose starting point x=h to determine whether sine or cosine and whether a>0 or a<0. Find the Rule 1 − −3 𝐴= =2 2 k= 1+ −3 2 = −1 2𝜋 2𝜋 𝑏= = =1 𝑃 2𝜋 Choose to start at max: h=π/2, +cosine 𝜋 𝑦 = 2 cos(𝑥 − ) − 1 2 Rule from TOV 𝜃 0 Volts 0 𝜋 4 77.8 Sketch to Visualize 𝜋 2 110 3𝜋 4 77.8 𝜋 0 3𝜋 2 −110 min 𝑓 = −110 𝑃 = 2𝜋 110 − (−110) 𝐴= 2 𝐴 = 110 = |𝑎| |𝑏| = Choosing h=0, makes +sine 𝑓 𝑥 = 110 sin 𝑥 0 Which means What do you know? max 𝑓 = 110 2𝜋 k= 2𝜋 =1 𝑃 110+ −110 2 =0 Rules from Words Find the rule of the sinusoidal function that • A = (ymax – ymin )/2 = 2 has a maximum at point A 𝜋 𝐴 , 10 4 • has a minimum at point B P = (5π/4 – π/4)×2 = 2 π b = 2 π/P = 1 5𝜋 𝐵 ,6 4 • k = (ymax + ymin )/2 = 8 h = π/4 (choosing +cosine) Only passes over the sinusoidal axis once between points A and B f(x) = 2cos(x - π/4) + 8 Rules from Words Find the rule of the sinusoidal function that • A = (ymax – ymin )/2 = 7 has a minimum at point A 3𝜋 𝐴 , −5 4 • k = (ymax + ymin )/2 = 2 P = (9π/4 – 3 π /4)×2 = 3 π b = 2 π/P = 2/3 has a maximum at point B h = 3π/4 (choosing -cosine) 9𝜋 𝐵 ,9 4 • Only passes over the sinusoidal axis once between points A and B f(x) = -7cos(2/3(x - 3π/4)) + 2 Rules from Words Find the rule of the sinusoidal function that • A = (ymax – ymin )/2 = 7 has a minimum at point A 3𝜋 𝐴 , −5 4 • k = (ymax + ymin )/2 = 2 P = (9π/4 – 3/4)×2 = 3 π b = 2 π/P = 2/3 has a maximum at point B h = 3π/2 (choosing +sine) 9𝜋 𝐵 ,9 4 • Only passes over the sinusoidal axis once between points A and B f(x) = 7sin(2/3(x - 3π/2)) + 2 HW Visions 2 Page 109: #6 Chapter 3: Trigonometry Lessons 3.08D Finding Rule of Sinusoidal Functions CONTINUED Vertex Period Practice Word Problems A = 3.5 ymin = -1 -> k = 2.5 f = 3 per min -> P = 1/3 min P = 20 s -> b = 2π/20 = π/10 At t=0, y = -1 (at min) -> -cosine f(x) = -3.5cos(π/10 * x)+2.5 f(7) = -3.5cos(π/10 * 7) +2.5 f(7) = 4.56 m