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

2. 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

3. (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

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

5. 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

6. 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

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

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

9. 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

10. (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

11. (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

12. (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

13. (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

14. 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
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Pre-Calculus

15. More Graphing with transformations
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Pre-Calculus

16. More Graphing with transformations
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Pre-Calculus

17. Write the equation from the graph
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Pre-Calculus

18. Write the equation from the graph
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Pre-Calculus

19. Write the equation from the graph
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Pre-Calculus

20. Write the equation from the graph
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Pre-Calculus

21. Write the equation from the graph
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Pre-Calculus

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

23. 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

24. 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