write the equation of the trigonometric function shown in the graph. use a positive coefficient on cosine…

Explanation:

Step1: Determine the amplitude

The general form of a cosine - function is $y = A\cos(B(x - C))+D$. The amplitude $|A|$ is half the distance between the maximum and minimum values of the function. From the graph, the maximum value is approximately $y = 3$ and the minimum value is approximately $y=- 9$. So, $A=\frac{3 - (-9)}{2}=\frac{3 + 9}{2}=6$.

Step2: Determine the period

The period of a cosine function $y = A\cos(B(x - C))+D$ is $T=\frac{2\pi}{|B|}$. If we assume the standard period of the cosine - function is $2\pi$ and there is no horizontal stretching or shrinking visible in the graph (a full - cycle seems to be over an interval of length $2\pi$), then $B = 1$.

Step3: Determine the phase - shift

The phase - shift of the cosine function is given by $C$. The cosine function $y=\cos x$ has a maximum at $x = 0$. For the given function, the maximum occurs at $x = 0$. So, the phase - shift $C = 0$.

Step4: Determine the vertical shift

The vertical shift $D$ is the mid - value between the maximum and minimum of the function. $D=\frac{3+( - 9)}{2}=\frac{3 - 9}{2}=-3$.

Answer:

$y = 6\cos(x)-3$