Question

What cosine function represents an amplitude of 2, a period of 2π, a horizontal shift of π, and a vertical shift of −1?

A. y=2cos(x−π)−1
B. y=−2cos(x+π)−1
C. y=2cos(x+π)−1
D. y=−2cos(x−π)−1

Answer 1

The correct cosine function is y = 2cos(x - π) - 1, matching option A, as it fulfills the given conditions of an amplitude of 2, a period of 2π, a horizontal shift of π, and a vertical shift of -1.

Step-by-step explanation:

The cosine function that represents an amplitude of 2, a period of 2π, a horizontal shift of π, and a vertical shift of −1 is given by the standard form of a cosine function y = A cos(B(x - C)) + D, where A is the amplitude, B is related to the period by the formula B = 2π / period, C is the horizontal shift (also known as the phase shift), and D is the vertical shift.

For the given conditions, the amplitude A is 2, the period suggests that B is 1 (since B = 2π / period and the period is 2π), the horizontal shift C is π (taking the cosine function to the right), and the vertical shift D is −1. The resulting function is therefore:

y = 2cos(x - π) - 1

This matches option A: y = 2cos(x - π) - 1.