Question

select the graph of $y = 2\cos(4x) - 1$

Explanation:

Step1: Analyze the function \( y = 2\cos(4x) - 1 \)

• Amplitude: The amplitude of \( A\cos(Bx)+C \) is \( |A| \). Here, \( A = 2 \), so amplitude is \( 2 \).
• Vertical Shift: The vertical shift is \( C=-1 \), so the midline is \( y=-1 \).
• Period: The period of \( \cos(Bx) \) is \( \frac{2\pi}{|B|} \). Here, \( B = 4 \), so period is \( \frac{2\pi}{4}=\frac{\pi}{2} \).

Step2: Evaluate the options

• Option A: Check amplitude, vertical shift, and period. The midline seems around \( y = -1 \) (vertical shift -1), amplitude 2 (from midline to peak is 2), and period \( \frac{\pi}{2} \) (distance between peaks is \( \frac{\pi}{2} \)) – matches.
• Option B: Midline is \( y = 1 \) (vertical shift 1), which doesn't match \( C=-1 \). Eliminate.
• Option C: Amplitude seems 1 (from midline to peak is 1), doesn't match \( A = 2 \). Eliminate.
• Option D: Period seems \( \pi \) (distance between peaks is \( \pi \)), doesn't match period \( \frac{\pi}{2} \). Eliminate.

Answer:

A. The graph labeled A