Question

select the correct answer. consider the function $y = \frac{1}{2} \cos(2x) - 1$. which statement is true about the function? \\(\bigcirc\\) a. the domain of the function is $-1.5, 0.5$. \\(\bigcirc\\) b. the amplitude of the function is 1. \\(\bigcirc\\) c. the period of the function is $2\pi$. \\(\bigcirc\\) d. the midline of the function is $y = -1$.

Explanation:

Step1: Analyze Option A (Domain)

The function \( y = \frac{1}{2}\cos(2x)-1 \) is a cosine function. Cosine functions have a domain of all real numbers (\( (-\infty, \infty) \)), not \([-1.5, 0.5]\). So A is false.

Step2: Analyze Option B (Amplitude)

The general form of a cosine function is \( y = A\cos(Bx - C)+D \), where \( |A| \) is the amplitude. Here, \( A = \frac{1}{2} \), so amplitude is \( \frac{1}{2} \), not 1. So B is false.

Step3: Analyze Option C (Period)

The period of \( y = A\cos(Bx - C)+D \) is \( \frac{2\pi}{|B|} \). Here, \( B = 2 \), so period is \( \frac{2\pi}{2}=\pi \), not \( 2\pi \). So C is false.

Step4: Analyze Option D (Midline)

The midline of \( y = A\cos(Bx - C)+D \) is \( y = D \). Here, \( D = -1 \), so midline is \( y = -1 \). This matches.

Answer:

D. The midline of the function is \( y = -1 \).