g is a trigonometric function of the form $g(x)=asin(bx + c)+d$. below is the graph $g(x)$. the function…

Explanation:

Step1: Find the value of (d)

The mid - line of the function (y = a\sin(bx + c)+d) is (y = d). Since the function intersects its mid - line at ((-1,6)), then (d = 6).

Step2: Find the value of (a)

The amplitude (a) is the distance between the mid - line and the maximum or minimum value of the function. The mid - line is (y = 6) and the minimum value is (y = 3). So (a=|6 - 3|=3). Since the function has a minimum point before the mid - line intersection point in the given part of the graph, (a = 3).

Step3: Find the value of (b)

The period (T) of a sine function (y=a\sin(bx + c)+d) can be found from the horizontal distance between a mid - line intersection and a minimum point. The horizontal distance from the mid - line point ((-1,6)) to the minimum point ((-3.5,3)) is (\frac{T}{4}). So (\frac{T}{4}=|-1-(-3.5)| = 2.5), then (T = 10). Since (T=\frac{2\pi}{|b|}) and for a sine function with the standard form we can assume (b>0), then (10=\frac{2\pi}{b}), and (b=\frac{\pi}{5}).

Step4: Find the value of (c)

We know that the function (g(x)=3\sin(\frac{\pi}{5}x + c)+6) passes through the point ((-1,6)). Substitute (x=-1) and (y = 6) into the function: [6=3\sin\left(\frac{\pi}{5}\times(-1)+c\right)+6] [0 = 3\sin\left(-\frac{\pi}{5}+c\right)] (\sin\left(-\frac{\pi}{5}+c\right)=0), so (-\frac{\pi}{5}+c = k\pi), (k\in\mathbb{Z}). Let (k = 0), then (c=\frac{\pi}{5}).

Answer:

(g(x)=3\sin\left(\frac{\pi}{5}x+\frac{\pi}{5}\right)+6)