Question

Given the cosine function f(x)=Acos(Bx−C)+D, where A, B, C, and D are constants, which of the following statements is correct regarding the function's characteristics?

A) The amplitude is A, the period is 2/2π/B, the phase shift is /C/B, and the vertical shift is D.
B) The amplitude is B, the period is /2π/A, the phase shift is /C/B, and the vertical shift is D.
C) The amplitude is A, the period is 2/2π/B, the phase shift is /C/B, and the vertical shift is −−D.
D) The amplitude is A, the period is 2/2π/B, the phase shift is −/−C/B, and the vertical shift is D.

Answer 1

The amplitude of the cosine function is A, the period is 2π/B, the phase shift is -C/B, and the vertical shift is D. Among the given options, option D correctly describes the characteristics of the cosine function with constants A, B, C, and D.

Step-by-step explanation:

When analyzing the cosine function f(x) = Acos(Bx - C) + D, the correct statements regarding the function's characteristics are as follows:

• The amplitude is A.
• The period is 2π/B. The 2π in the period comes from the periodic nature of the cosine function, which repeats every 2π radians. In this function f(x), the variable B affects the frequency of the oscillation, and thus the period is 2π divided by B.
• The phase shift is -C/B. A negative value in the function suggests that the graph is shifted to the right by C/B.
• The vertical shift is D, which moves the entire function up or down on the coordinate plane.

Therefore, with reference to the given cosine function, the corresponding statements regarding amplitude, period, phase shift, and vertical shift are most accurately reflected in option D. The amplitude is A, the period is 2π/B, the phase shift is -C/B, and the vertical shift is D.