question 13\nnext question\nwrite the equation of the trigonometric function shown in the graph.\n( y=square…

Answer:

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

Explanation:

Step1: Find the amplitude

The amplitude $A$ of a cosine function $y = A\cos(Bx)+C$ is half the distance between the maximum and minimum values. From the graph, the maximum value of the dashed - line (cosine - like) function is $y = 3$ and the minimum is $y=-1$. So, $A=\frac{3 - (- 1)}{2}=\frac{4}{2}=2$.

Step2: Find the vertical shift

The vertical shift $C$ is the mid - value between the maximum and minimum. $C=\frac{3+( - 1)}{2}=\frac{2}{2}=1$.

Step3: Find the period and then the value of $B$

The general formula for the period of a cosine function $y = A\cos(Bx)+C$ is $T=\frac{2\pi}{|B|}$. The period of the cosine function (dashed line) from the graph: If we consider two consecutive peaks (or troughs), say from $x = 0$ to $x = 4\pi$. So, $T = 4\pi$. Since $T=\frac{2\pi}{|B|}$ and $T = 4\pi$, then $4\pi=\frac{2\pi}{|B|}$. Solving for $B$: [ \begin{align*} 4\pi|B|&=2\pi\ |B|&=\frac{2\pi}{4\pi}=\frac{1}{2} \end{align*} ] We can take $B=\frac{1}{2}$ (since the function is a cosine function and we are not considering reflections for the basic form in the given template $y = A\cos(Bx)+C$).

So the equation of the trigonometric function is $y = 2\cos\left(\frac{1}{2}x\right)+1$.