What is the general equation of a cosine function with an amplitude of 3, a period of [tex]$4 \pi$[/tex], and a horizontal shift of [tex]$-\pi$[/tex]?
A. [tex]$y=4 \pi \cos (3(x-\pi))$[/tex]
B. [tex]$y=3 \cos (4 \pi(x+\pi))$[/tex]
C. [tex]$y=3 \cos (0.5(x+\pi))$[/tex]
D. [tex]$y=4 \pi \cos (0.2(x+\pi))$[/tex]
Answer (2)
The general form of a cosine function is y = A cos ( B ( x − C )) + D .
Identify the given parameters: amplitude A = 3 , period = 4 π , horizontal shift C = − π , and vertical shift D = 0 .
Calculate B using the period formula: B = 4 π 2 π = 0.5 .
Plug the values into the general form to get the equation: y = 3 cos ( 0.5 ( x + π )) .
y = 3 cos ( 0.5 ( x + π ))
Explanation
Problem Analysis We are given a cosine function with an amplitude of 3, a period of 4 π , and a horizontal shift of − π . We need to find the general equation of this cosine function.
General Form of Cosine Function The general form of a cosine function is given by: y = A cos ( B ( x − C )) + D where:
A is the amplitude
B 2 π is the period
C is the horizontal shift
D is the vertical shift
Identifying Parameters In our case, we have:
Amplitude A = 3
Period = 4 π
Horizontal shift C = − π
Since no vertical shift is mentioned, we assume D = 0 .
Calculating B We need to find the value of B using the period: 4 π = B 2 π Solving for B :
B = 4 π 2 π = 2 1 = 0.5
Final Equation Now we can plug in the values of A , B , C , and D into the general form: y = 3 cos ( 0.5 ( x − ( − π ))) + 0 y = 3 cos ( 0.5 ( x + π ))
Conclusion Therefore, the general equation of the cosine function is: y = 3 cos ( 0.5 ( x + π ))
Examples
Cosine functions are incredibly useful in modeling periodic phenomena, such as sound waves, light waves, and even the cyclical patterns in financial markets. For instance, understanding the properties of a cosine function can help predict the high and low tides at a beach, optimize the acoustics in a concert hall, or analyze the seasonal trends in stock prices. By adjusting the amplitude, period, and phase shift, we can tailor the cosine function to fit a wide range of real-world scenarios, making it an indispensable tool in science and engineering.
The general equation of the cosine function with an amplitude of 3, a period of 4 π , and a horizontal shift of − π is y = 3 cos ( 0.5 ( x + π )) . The correct multiple-choice option is C. This equation captures the described properties of the cosine function accordingly.
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