write the equation of the trigonometric function shown in the graph. try whole numbers or unit fractions for…

Explanation:

Step1: Determine the amplitude

The amplitude $A$ of a cosine - function $y = A\cos(Bx)+C$ is half the distance between the maximum and minimum values. The maximum value of the function is $y = 5$ and the minimum value is $y=- 5$. So, $A=\frac{5 - (-5)}{2}=5$.

Step2: Determine the period

The standard period of $y = \cos(x)$ is $2\pi$. Let the period of the given function be $T$. From the graph, we can see that one complete cycle occurs over an interval of length $3\pi$. The formula for the period of $y=\cos(Bx)$ is $T=\frac{2\pi}{B}$. Since $T = 3\pi$, we have $3\pi=\frac{2\pi}{B}$, and solving for $B$ gives $B=\frac{2}{3}$.

Step3: Determine the vertical shift

The mid - line of the function is $y = 0$ (since the function oscillates symmetrically about the $x$ - axis), so the vertical shift $C = 0$.

Answer:

$y = 5\cos\left(\frac{2}{3}x\right)+0$ or simply $y = 5\cos\left(\frac{2}{3}x\right)$