find an equation for the cosine graph, f(x):\nwrite your answer in the form f(x) = a cos(bx + c)+d, where a…

Explanation:

Step1: Determine the amplitude A

The amplitude is half the distance between the maximum and minimum values. The maximum value is 3 and the minimum is 1. So, $A=\frac{3 - 1}{2}=1$.

Step2: Determine the vertical - shift D

The vertical - shift is the average of the maximum and minimum values. So, $D=\frac{3 + 1}{2}=2$.

Step3: Determine the period and B

The period of the cosine function is the distance between two consecutive peaks. Here, the period $T = 4\pi$. Since $T=\frac{2\pi}{B}$, then $4\pi=\frac{2\pi}{B}$, and solving for B gives $B=\frac{2\pi}{4\pi}=\frac{1}{2}$.

Step4: Determine the phase - shift C

The graph is not shifted horizontally (the cosine function starts at its maximum at $x = 0$), so $C = 0$.

Answer:

$f(x)=1\cos(\frac{1}{2}x+0)+2=\cos(\frac{1}{2}x)+2$