find an equation for the cosine graph, f(x):

Explanation:

Step1: Determine the amplitude

The amplitude $A$ is half the vertical distance between the maximum and minimum values. The maximum value is $2$ and the minimum is $- 2$. So $A=\frac{2 - (-2)}{2}=2$.

Step2: Determine the vertical shift

The mid - line of the graph is at $y = 0$. The general form of a cosine function is $y=A\cos(Bx - C)+D$, where $D$ is the vertical shift. Here $D = 0$.

Step3: Determine the period

The period $P$ of the cosine function is the horizontal distance between two consecutive maxima. From the graph, $P = 2\pi$. The formula for the period of $y = A\cos(Bx - C)+D$ is $P=\frac{2\pi}{|B|}$. Since $P = 2\pi$, then $|B| = 1$. We can assume $B = 1$ for a non - horizontally reflected cosine function.

Step4: Determine the phase shift

The graph passes through the point $(0,2)$ which is the maximum of the cosine function, so the phase shift $C = 0$.

Answer:

$f(x)=2\cos(x)$