the following graph shows at least one complete cycle of the graph of an equation containing a trigonometric…

Explanation:

Step1: Identify the general form

The general form of a sinusoidal - function is $y = A\sin(B(x - C))+D$ or $y = A\cos(B(x - C))+D$. The graph passes through the origin, so we can start with the sine - function form $y = A\sin(Bx)+D$.

Step2: Determine the amplitude $A$

The amplitude $A$ is half the distance between the maximum and minimum values of the function. The maximum value is $y = 5$ and the minimum value is $y=-5$. So, $A=\frac{5 - (- 5)}{2}=5$.

Step3: Determine the period $T$ and $B$

The period $T$ is the length of one complete cycle. From the graph, $T = 2$. The formula for the period of a sine function is $T=\frac{2\pi}{B}$. Since $T = 2$, we have $2=\frac{2\pi}{B}$, and solving for $B$ gives $B=\pi$.

Step4: Determine the vertical shift $D$

The mid - line of the function is $y = 0$ (since the graph is symmetric about the $x$ - axis), so $D = 0$. And since there is no horizontal shift ($C = 0$ as the graph starts at the origin for the sine - like pattern), the equation of the function is $y = 5\sin(\pi x)$.

Answer:

$y = 5\sin(\pi x)$