determine the simplest form of an equation for the graph. choose b > 0, and include no phase shifts…

Explanation:

Step1: Identify the function type

The graph appears to be a sine - like function of the form $y = A\sin(bx)+k$.

Step2: Find the vertical shift $k$

The mid - line of the graph is $y = 1$. So, $k = 1$.

Step3: Find the amplitude $A$

The distance from the mid - line to the maximum (or minimum) is $A$. The maximum value is $y = 2$ and the mid - line is $y = 1$, so $A=1$.

Step4: Find the period $T$

The period $T$ is the distance between two consecutive maximums or minimums. From the graph, the period $T=\frac{\pi}{6}-\left(-\frac{\pi}{6}\right)=\frac{\pi}{3}$.

Step5: Calculate $b$

We know that the formula for the period of $y = A\sin(bx)+k$ is $T=\frac{2\pi}{b}$. Since $T = \frac{\pi}{3}$, we have $\frac{\pi}{3}=\frac{2\pi}{b}$. Solving for $b$ gives $b = 6$.

Answer:

$y=\sin(6x)+1$