6-10. free response. show all work. (50 points)\n6. (10pts) find the equation of the graphed function, given…

Explanation:

Step1: Determine the vertical shift

The mid - line of the parent function $y = \sin(x)$ is $y = 0$. The mid - line of the given graph is $y = 2$. So the vertical shift $d=2$.

Step2: Determine the amplitude

The amplitude $A$ of a sinusoidal function $y = A\sin(Bx - C)+D$ is the distance from the mid - line to the maximum or minimum. The maximum value of the given function is $3$ and the mid - line is $y = 2$, so $A=3 - 2=1$.

Step3: Determine the period

The period of the parent function $y=\sin(x)$ is $2\pi$. The period of the given function is $\frac{2\pi}{3}$. We know that the period formula is $T=\frac{2\pi}{B}$, and since $T = \frac{2\pi}{3}$, then $\frac{2\pi}{B}=\frac{2\pi}{3}$, so $B = 3$.

Step4: Determine the phase shift

When $x = 0$, the function passes through the mid - line value, so the phase shift $C = 0$. The general form of a sinusoidal function is $y=A\sin(Bx - C)+D$. Substituting $A = 1$, $B = 3$, $C = 0$ and $D = 2$ into the formula, we get $y=\sin(3x)+2$.

Answer:

$y=\sin(3x)+2$