which of the following is an equation for the graph?\na. $2cos(x + \frac{pi}{2})+1$ b. $2cos(x +…

Explanation:

Step1: Recall general cosine - function form

The general form of a cosine function is $y = A\cos(B(x - C))+D$, where $A$ is the amplitude, $B$ affects the period ($T=\frac{2\pi}{|B|}$), $C$ is the phase - shift, and $D$ is the vertical shift.

Step2: Determine the amplitude

The amplitude $A$ is half of the vertical distance between the maximum and minimum values of the function. The maximum value of the given graph is $y = 1$ and the minimum value is $y=-3$. So, $A=\frac{1 - (-3)}{2}=\frac{4}{2}=2$.

Step3: Determine the vertical shift

The vertical shift $D$ is the mid - value between the maximum and minimum values. $D=\frac{1+( - 3)}{2}=\frac{-2}{2}=-1$.

Step4: Determine the phase - shift

The standard cosine function $y = \cos x$ has a maximum at $x = 0$. The given cosine - like function has a maximum at $x=-\frac{\pi}{2}$. For the function $y = A\cos(B(x - C))+D$, when $B = 1$, the phase - shift $C$ is such that the function is shifted to the left by $\frac{\pi}{2}$ units. So, the function is of the form $y=2\cos(x+\frac{\pi}{2})-1$.

Answer:

B. $2\cos(x+\frac{\pi}{2})-1$