Lesson 30 – Trigonometric Functions & Periodic Phenomenon Pre-Calculus 11/11/2018 Pre-Calculus

Lesson Objectives 1. Relate the features of sinusoidal curves to modeling periodic phenomenon 2. Transformations of sinusoidal functions and their features 11/11/2018 Pre-Calculus

(A) Key Terms Define the following key terms that relate to trigonometric functions: (a) period (b) amplitude (c) axis of the curve (or equilibrium axis) 11/11/2018 Pre-Calculus

(A) Key Terms 11/11/2018 Pre-Calculus

The General Sinusoidal Equation In the equation f(x) = asin(k(x+c)) + d, explain what: a represents? k represents? c represents? d represents? 11/11/2018 Pre-Calculus

Contextual Analysis: DATA SET Follow this link for the graph of the data. Adjust the sliders to fit the data. What are each variable affecting? 11/11/2018 Pre-Calculus

(C) Modeling Periodic Phenomenon & Transformed Sinusoidal Curves 11/11/2018 Pre-Calculus

(C) Modeling Periodic Phenomenon & Transformed Sinusoidal Curves 11/11/2018 Pre-Calculus

Examples to Develop Sketch  2 cycles Analyze  (D, R, max/min, roots, period, amplitude, axis of curve) (a) the function (b) the function (c) the function (d) the function 11/11/2018 Pre-Calculus

(C) Modeling Periodic Phenomenon & Transformed Sinusoidal Curves A spring bounces up and down according to the model d(t) = 0.5cos(2t), where d is the displacement in centimetres from the rest position and t is the time in seconds. The model does not consider the effects of gravity. (a) Make a table for 0 ≤ t ≤ 9, using 0.5-s intervals. (b) Draw the graph. (c) Explain why the function models periodic behaviour. (d) What is the relationship between the amplitude of the function and the displacement of the spring from its rest position? (e) What is the period and what does it represent in the context of this question? (f) What is the amplitude and what does it represent in the context of this question? 11/11/2018 Pre-Calculus

(E) Combining Transformations We continue our investigation by graphing some other functions in which we have combined our transformations (i) Graph and analyze  identify transformations and state how the key features have changed (ii) Graph and analyze  identify transformations and state how the key features have changed (iii) Graph and analyze  identify transformations and state how the key features have changed 11/11/2018 Pre-Calculus

(B) Writing Sinusoidal Equations ex 1. Given the equation y = 2sin3(x - 60°) + 1, determine the new amplitude, period, phase shift and equation of the axis of the curve. Amplitude is obviously 2 Period is 2π/3 or 360°/3 = 120° The equation of the equilibrium axis is y = 1 The phase shift is 60° to the right 11/11/2018 Pre-Calculus

(B) Writing Sinusoidal Equations ex 2. Given a cosine curve with an amplitude of 2, a period of 180°, an equilibrium axis at y = -3 and a phase shift of 45° right, write its equation. So the equation is y = 2 cos [2(x - 45°)] – 3 Recall that the k value is determined by the equation period = 2π/k or k = 2π/period If working in degrees, the equation is modified to period = 360°/k or k = 360°/period 11/11/2018 Pre-Calculus

Writing sinusoidal equations Given a tangent curve with amplitude of 4, period of 2π/3, reflected across the x-axis, shifted up 5, and a phase shift of π/4 Given a sine curve with amplitude of ½, period of 270°, phase shift of 60° Given secant curve with amplitude of 3, period of 6π, shifted down 2. Given cosecant curve with amplitude of 4, reflected across the x-axis, period of 4 11/11/2018 Pre-Calculus

More Graphing with transformations 11/11/2018 Pre-Calculus

More Graphing with transformations 11/11/2018 Pre-Calculus

Write the equation from the graph 11/11/2018 Pre-Calculus

Write the equation from the graph 11/11/2018 Pre-Calculus

Write the equation from the graph 11/11/2018 Pre-Calculus

Write the equation from the graph 11/11/2018 Pre-Calculus

Write the equation from the graph 11/11/2018 Pre-Calculus

Write the equation from the graph 11/11/2018 Pre-Calculus

Applications of Sinusoidal Fcns Write the equation that can be used to model the following relationship between a riders height (in m above the ground) and time spent on the ride, in seconds since the rider started the ride. NOTE: assume the rider got on the ride when the wheel was at its lowest height. A Ferris wheel with a radius of 7 m, whose axle is 9 m above the ground and that rotates once every 40 seconds. Use your equation to predict: (a) the rider height after 15 seconds (b) at what time(s) the rider is 14 m above the ground 11/11/2018 Pre-Calculus

Applications of Sinusoidal Fcns The average monthly temperature, T(t), in degrees Celsius in Kingston, Ontario, can be modeled by the function below, where t represents the number of months (for t = 1, the month is January) (a) Determine the period and state its significance in this context (b) Determine the equation of the axis of the curve and state its significance in this context (c) Determine the amplitude of the curve and state its significance in this context (d) Graph the function on your graphing calculator (e) Evaluate T(30) and explain what your answer means in this context (f) Solve T(t) = 12 and explain what your answer means in this context 11/11/2018 Pre-Calculus