question 16\nnext question\nwrite the equation of the trigonometric function shown in the graph.\ntry\n$y =…

Answer:

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

Explanation:

Step1: Recall the general form of cosine function

The general form is (y = A\cos(Bx)+C), where (A) is amplitude, (B) affects the period ((T=\frac{2\pi}{|B|})), and (C) is the vertical shift.

Step2: Determine the amplitude (A)

The amplitude (A) is the distance from the mid - line to the maximum (or minimum) value. The mid - line is (y = 1) (since the graph oscillates around (y = 1)), and the distance from (y = 1) to (y=3) (or (y=-1)) is (|A| = 2).

Step3: Determine the vertical shift (C)

The mid - line of the cosine function is (y = C). From the graph, the mid - line is (y = 1), so (C = 1).

Step4: Determine the period (T) and find (B)

The period (T) is the length of one full cycle. From the graph, the period (T=3\pi). Using the formula (T=\frac{2\pi}{|B|}), we substitute (T = 3\pi) into it: (3\pi=\frac{2\pi}{|B|}). Solving for (B), we get (|B|=\frac{2}{3}). Since there is no reflection (the graph starts at the maximum for a cosine - type function, and (A>0)), we can take (B=\frac{2}{3}).

Substituting (A = 2), (B=\frac{2}{3}), and (C = 1) into the general form (y=A\cos(Bx)+C), we get (y = 2\cos\left(\frac{2}{3}x\right)+1).