the function graphed is of the form y = a sin bx or y = a cos bx, where b > 0. determine the equation of the…

Explanation:

Step1: Determine the amplitude a

The amplitude is the maximum distance from the mid - line of the graph to its peak or trough. The mid - line is $y = 0$. The maximum value of the function is $y = 5$ and the minimum is $y=-5$. So, $a = 5$.

Step2: Determine the period and find b

The period $T$ of a function $y=a\sin(bx)$ or $y = a\cos(bx)$ is given by $T=\frac{2\pi}{b}$. The graph goes from $x = 0$ to $x = 2\pi$ and completes one - half of a cycle. So the full period $T = 4\pi$. Since $T=\frac{2\pi}{b}$ and $T = 4\pi$, we have $4\pi=\frac{2\pi}{b}$. Solving for $b$ gives $b=\frac{1}{2}$.

Step3: Determine the type of the trigonometric function

The graph passes through the point $(0, - 5)$. For $y=a\sin(bx)$, when $x = 0$, $y=a\sin(0)=0$. For $y=a\cos(bx)$, when $x = 0$, $y=a\cos(0)=a$. Since the graph passes through $(0,-5)$ and $a = 5$, the function is $y=-a\cos(bx)$.

Answer:

$y=-5\cos\left(\frac{1}{2}x\right)$