Question

periodic functions
periodic function flexibility
using the given graph, write an equation for a cosine function.
(1 point)
$y = square$

Explanation:

Step1: Determine the amplitude

The amplitude \( A \) of a cosine function \( y = A\cos(Bx - C)+D \) is half the distance between the maximum and minimum values. The maximum value \( y_{max}=4 \) and the minimum value \( y_{min}=0 \). So, \( A=\frac{y_{max}-y_{min}}{2}=\frac{4 - 0}{2}=2 \).

Step2: Determine the vertical shift

The vertical shift \( D \) is the mid - line of the function. \( D=\frac{y_{max}+y_{min}}{2}=\frac{4 + 0}{2}=2 \).

Step3: Determine the period and \( B \)

The period \( P \) of the function. From the graph, the period \( P = 2\pi \). The formula for the period of a cosine function \( y = A\cos(Bx - C)+D \) is \( P=\frac{2\pi}{|B|} \). Since \( P = 2\pi \), then \( \frac{2\pi}{|B|}=2\pi \), so \( B = 1 \).

Step4: Determine the phase shift

Since there is no horizontal shift (the graph has no left - right movement from the standard \( y=\cos(x) \) graph when considering the key points), \( C = 0 \).

Answer:

\( y=2\cos(x)+2 \)