### Determine the period, amplitude and phase shift for each given function

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A+ SOLUTIONS FOR THE FOLLOWING QUESTIONS

**Trigonometric Functions and Graphs**

Applications of Trigonometric Functions

Trig Functions & Graphs Unit Assignment

The following questions are to be answered with full solutions. Be sure to focus on proper mathematical form, including:

Assessment OF Learning: Trig Functions & Graphs Unit Assignment

1.Determine the period, amplitude and phase shift for each given function: (12 marks)

a.y = -4 cos 3x + 5

b.y = - 10

c.y = -0.38 tan

d.

2.For the function between x = 0 and x = 2 : (6 marks)

a.For what value(s) of x does y have its maximum value?

b.For what value(s) of x does y have its minimum value?

c.For what value(s) of x does y = 1?

3.Find the following functions (constants should be exact if possible; otherwise to two decimal places): (12 marks)

a.of the form y = sin x + c that passes through (225, 3)

b.of the form y = cos x + c that passes through (30, -2)

c.of the form y = a tan x that passes through

d.of the form y = sin(x-d) that passes through (270, )

e.of the form y = cos(x-d) that passes through

f.of the form y = tan kx that passes through (290, -1)

4.Construct the equations of the following trigonometric functions: (6 marks)

a.A sine function with amplitude 2, period , phase shift /3 right

b.A tangent function with a reflection in the y-axis, period , translation up 5 units

c.A cosine function with period 270, translation down 50 units, reflection in the x-axis

5.The tide in a local costal community can be modeled using a sine function. Starting at noon, the tide is at its "average" height of 3 metres measured on a pole located off of the shore. 5 hours later is high tide with the tide at a height of 5 metres measured at the same pole. 15 hours after noon is low tide with the tide at a height of 1 metre measured at the same pole. Use this information to model the tide motion using a sine function. Show all work. (4 marks)

6.A mass is supported by a spring so that it is at rest 0.5 m above a tabletop. The mass is pulled down 0.4 m and released at time t = 0. It then moves up through its rest position, stops and drops again, creating a periodic up and down motion that can be modeled using a trigonometric function. It takes 1.2 seconds to return to the lowest position each time.

a.Sketch a graph showing the height of the mass above the tabletop as a function of time for the first 3 seconds of its motion. (4 marks)

b.Write an equation for the function in part a). (4 marks)

c.Use your equation to determine the height of the mass above the tabletop after (3 marks)

i.0.3 seconds

ii.0.7 seconds

iii.2.2 seconds

d.What assumption(s) must be made for this function to be valid? Explain. (2 marks)

7.Last summer Cooltown, Ontario, experienced several consecutive days of warm temperatures. Below are the temperatures measured for a 24 hour period beginning at midnight and ending the next midnight.020021.5110029.5200029.4

TimeTemp (C)TimeTemp (C)TimeTemp (C)

000023.3090026.3180031.8

010022.2100027.8190030.5

030021.1120030.6210027.8

040021.1130031.8220026.2

050021.5140032.5230024.7

060022.2150032.9240023.4

070023.5160032.9

080024.9170032.6

a.Plot the above data. Note: start your vertical scale at a value which takes into consideration the range of the data. For the horizontal scale start with 0 for midnight, then 1 for 1 hour after midnight (0100) and so on. (4 marks)

b.Draw a smooth curve through the points you have plotted. (1 mark)

c.Based on the information from the smooth curve, find an equation for the temperature T in terms of the time t in the form T(t) = a sin(kt ? c) + d where a, k, c and d are constants. (4 marks)

d.From your equation predict the temperature at 4:30 pm. (1 mark)

e.From your equation, when during the day would the temperature be 30C? (2 marks)

8.Explain the terms domain and range. Explain how restrictions to these arise for the trigonometric functions. (3 marks)

9.For what values of d do the functions y = cos x and y = cos(x d) have the same values? Explain. (2 marks)

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