Explanation Result used: The functions y = a sin ( b x + c ) or y = a cos ( b x + c ) each has an amplitude of | a | , a period of 2 π b and displacement − c b . Calculation: Given that the function of the form y = 3 sin ( π x + c ) passes through ( − 0.25 , 0 ) and c has the smallest possible positive value. That is, c > 0 . Obtain the required function as follows: Substitute x = − 0.25 and y = 0 in y = 3 sin ( π x + c ) . y = 3 sin ( π x + c ) 0 = 3 sin ( − 0.25 π + c ) 0 = 3 sin ( − π 4 + c ) sin ( − π 4 + c ) = 0 Notice that, the value of the sine function is 0 for 0 , π , 2 π , 3 π ,.... That is, for n π , where n = 0 , 1 , 2 , … . Thus, − π 4 + c = n π . Simplify the above equation to obtain as, c = n π + π 4 . The smallest value of c such that c > 0 will be obtained when n = 0 . Substitute n = 0 in c = n π + π 4 . c = ( 0 ) π + π 4 = 0 + π 4 = π 4 Thus, the smallest value of c is π 4 . Substitute c = π 4 in y = 3 sin ( π x + c ) to obtain y = 3 sin ( π x + π 4 ) . Therefore, the function of the form y = 3 sin ( π x + c ) that passes through ( − 0.25 , 0 ) and c has the smallest possible positive value is y = 3 sin ( π x + π 4 ) _ . By the above mentioned result, the values of a , b and c for y = 3 sin ( π x + π 4 ) are 3 , π and π 4 , respectively. Since a = 3 , the maximum value will be 3 and the minimum value will be –3. The period of y = 3 sin ( π x + π 4 ) is obtained as follows: 2 π b = 2 π π ( ∵ b = π ) = 2 Thus, the period of y = 3 sin ( π x + π 4 ) is 2

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ISBN: 9780134508290

Author: Evans

Publisher: PEARSON CUSTOM PUB.(CONSIGNMENT)

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Chapter 10, Problem 67RE

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The function of the form y=3sin(πx+c) that passes through (−0.25,0) and c has the smallest possible positive value and graph the function.

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Chapter 10, Problem 66RE

Chapter 10, Problem 68RE

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Chapter 10 Solutions

EBK BASIC TECHNICAL MATHEMATICS

Ch. 10.1 - For the graph of y = −6 sin x, set up a table of...Ch. 10.1 - Prob. 1E Ch. 10.1 - Prob. 2E Ch. 10.1 - Prob. 3E Ch. 10.1 - In Exercises 3–6, complete the following table for...Ch. 10.1 - Prob. 5E Ch. 10.1 - Prob. 6E Ch. 10.1 - In Exercises 7–22, give the amplitude and sketch...Ch. 10.1 - Prob. 8E Ch. 10.1 - Prob. 9E

Ch. 10.1 - Prob. 10E Ch. 10.1 - Prob. 11E Ch. 10.1 - Prob. 12E Ch. 10.1 - In Exercises 7–22, give the amplitude and sketch...Ch. 10.1 - In Exercises 7–22, give the amplitude and sketch...Ch. 10.1 - Prob. 15E Ch. 10.1 - Prob. 16E Ch. 10.1 - Prob. 17E Ch. 10.1 - Prob. 18E Ch. 10.1 - Prob. 19E Ch. 10.1 - Prob. 20E Ch. 10.1 - Prob. 21E Ch. 10.1 - Prob. 22E Ch. 10.1 - Prob. 23E Ch. 10.1 - Prob. 24E Ch. 10.1 - Prob. 25E Ch. 10.1 - Prob. 26E Ch. 10.1 - Prob. 27E Ch. 10.1 - Prob. 28E Ch. 10.1 - Prob. 29E Ch. 10.1 - Prob. 30E Ch. 10.1 - Prob. 31E Ch. 10.1 - In Exercises 27–32, solve the given problems. 32....Ch. 10.1 - Prob. 33E Ch. 10.1 - Prob. 34E Ch. 10.1 - Prob. 35E Ch. 10.1 - Prob. 36E Ch. 10.1 - Prob. 37E Ch. 10.1 - Prob. 38E Ch. 10.1 - Prob. 39E Ch. 10.1 - Prob. 40E Ch. 10.2 - Prob. 1PE Ch. 10.2 - Prob. 2PE Ch. 10.2 - Prob. 3PE Ch. 10.2 - Prob. 1E Ch. 10.2 - Prob. 2E Ch. 10.2 - Prob. 3E Ch. 10.2 - In Exercises 3–22, find the amplitude and period...Ch. 10.2 - Prob. 5E Ch. 10.2 - In Exercises 3–22, find the amplitude and period...Ch. 10.2 - 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Prob. 43E Ch. 10.2 - Prob. 44E Ch. 10.2 - Prob. 45E Ch. 10.2 - Prob. 46E Ch. 10.3 - Prob. 1PE Ch. 10.3 - Prob. 2PE Ch. 10.3 - Prob. 1E Ch. 10.3 - Prob. 2E Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - Prob. 7E Ch. 10.3 - Prob. 8E Ch. 10.3 - Prob. 9E Ch. 10.3 - Prob. 10E Ch. 10.3 - Prob. 11E Ch. 10.3 - Prob. 12E Ch. 10.3 - Prob. 13E Ch. 10.3 - Prob. 14E Ch. 10.3 - Prob. 15E Ch. 10.3 - Prob. 16E Ch. 10.3 - Prob. 17E Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - Prob. 19E Ch. 10.3 - In Exercises 3–26, determine the amplitude,...Ch. 10.3 - Prob. 21E Ch. 10.3 - Prob. 22E Ch. 10.3 - Prob. 23E Ch. 10.3 - Prob. 24E Ch. 10.3 - Prob. 25E Ch. 10.3 - Prob. 26E Ch. 10.3 - Prob. 27E Ch. 10.3 - Prob. 28E Ch. 10.3 - Prob. 29E Ch. 10.3 - Prob. 30E Ch. 10.3 - Prob. 31E Ch. 10.3 - Prob. 32E Ch. 10.3 - 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The function of the form y = 3 sin ( π x + c ) that passes through ( − 0.25 , 0 ) and c has the smallest possible positive value and graph the function.

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The function of the form y=3sin(πx+c) that passes through (−0.25,0) and c has the smallest possible positive value and graph the function.

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Chapter 10 Solutions