Question

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

Explanation:

Step1: Analyze Option A

The function is a cosine function, \( y = \frac{1}{2}\cos(2x)-1 \). The domain of a cosine function is all real numbers (\( \mathbb{R} \)), not \([-1.5, 0.5]\) (which is a range - like interval). So A is false.

Step2: Analyze Option B

For a sinusoidal function of the form \( y = A\cos(Bx - C)+D \), the midline is \( y = D \). Here, \( D=-1 \), so the midline is \( y = -1 \). This matches the graph (the middle horizontal line around which the cosine wave oscillates).

Step3: Analyze Option C

The period of \( y = A\cos(Bx)+D \) 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

The amplitude of \( y = A\cos(Bx)+D \) is \( |A| \). Here, \( A=\frac{1}{2} \), so amplitude is \( \frac{1}{2} \), not 1. So D is false.

Answer:

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