by using a cosine function find a formula representation of the graph in the given figure. use ( ) around…

Explanation:

Step1: Determine the amplitude

The amplitude $A$ is half of the vertical distance between the maximum and minimum values of the function. The maximum value is $y = 2$ and the minimum value is $y = 0$, so $A=\frac{2 - 0}{2}=1$.

Step2: Determine the vertical - shift

The mid - line of the cosine function is $y = 1$. The general form of a cosine function is $y=A\cos(B(x - C))+D$, where $D$ is the vertical shift. So $D = 1$.

Step3: Determine the period

The period $T$ of the cosine function is the horizontal distance between two consecutive maxima or minima. From the graph, $T = 3$. The formula for the period of a cosine function $y = A\cos(B(x - C))+D$ is $T=\frac{2\pi}{B}$. Since $T = 3$, we have $3=\frac{2\pi}{B}$, so $B=\frac{2\pi}{3}$.

Step4: Determine the phase - shift

The cosine function $y=\cos x$ has a maximum at $x = 0$. The given cosine - like function has a maximum at $x = 0$, so the phase - shift $C = 0$.

Answer:

$y = 1+\cos(\frac{2\pi}{3}x)$