question 14\nnext question\nwrite the equation of the trigonometric function shown in the graph.\n( y=square…

Answer:

$y = 1\cos\left(\frac{2}{3}x\right)+0$

Explanation:

Step1: Determine the amplitude

The general form of a cosine function is $y = A\cos(Bx)+C$. The amplitude $A$ is the maximum distance from the mid - line. The mid - line of the function is $y = 0$ (since the graph oscillates equally above and below $y = 0$), and the maximum value of the function is $y = 1$. So, $A=1$.

Step2: Determine the vertical shift

The vertical shift $C$ is the mid - line of the function. Since the mid - line is $y = 0$, $C = 0$.

Step3: Determine the period and find $B$

The period of a cosine function $y=\cos(Bx)$ is $T=\frac{2\pi}{B}$. The standard period of $y = \cos(x)$ is $2\pi$. Looking at the graph, the period $T = 3\pi$. We know that $T=\frac{2\pi}{B}$, substituting $T = 3\pi$ into the formula: $$3\pi=\frac{2\pi}{B}$$ Cross - multiply gives $3\pi B=2\pi$. Divide both sides by $3\pi$: $B=\frac{2}{3}$.