write the equation of the trigonometric function shown in the graph.\n( y=square sin (square x)+square )

Answer:

$y = 3\sin\left(\frac{2}{3}x\right)+1$

Explanation:

Step1: Determine the amplitude

The amplitude (A) is half the distance between the maximum and minimum values. The maximum value of the dashed - line (sine - like) function is (y = 4) and the minimum is (y=-2). So, (A=\frac{4 - (- 2)}{2}=3).

Step2: Determine the vertical shift

The vertical shift (D) is the mid - value of the maximum and minimum. (D=\frac{4+( - 2)}{2}=1).

Step3: Determine the period

The period (P) of a sine function (y = A\sin(Bx)+D) is given by (P=\frac{2\pi}{B}). The period of the function (distance between two consecutive peaks) is (P = 3\pi). Using the formula (P=\frac{2\pi}{B}), we solve for (B): (3\pi=\frac{2\pi}{B}), so (B=\frac{2}{3}).

Since there is no horizontal shift (the graph has no phase - shift, as it can be considered as a standard sine - function transformation in terms of vertical shift, amplitude and period change), the equation of the trigonometric function is (y = 3\sin\left(\frac{2}{3}x\right)+1).