graph the following function. y = - 2+3 sin (x + \\frac{\\pi}{6}) use the graphing tool to graph the…

Explanation:

Step1: Identify the general form of a sinusoidal function

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$. For the given function $y=- 2 + 3\sin\left(x+\frac{\pi}{6}\right)$, we have $A = 3$, $B = 1$, $C=-\frac{\pi}{6}$, and $D=-2$.

Step2: Determine the amplitude

The amplitude is given by $|A|$. Since $A = 3$, the amplitude $|A|=3$.

Step3: Determine the period

The period of a sinusoidal function $y = A\sin(B(x - C))+D$ is $T=\frac{2\pi}{|B|}$. Since $B = 1$, the period $T = 2\pi$.

Step4: Determine the phase - shift

The phase - shift is given by $C$. Here, $C=-\frac{\pi}{6}$, which means a shift of $\frac{\pi}{6}$ units to the left.

Step5: Determine the vertical shift

The vertical shift is given by $D$. Here, $D=-2$, so the graph is shifted 2 units down.

Step6: Find key points

We know that for $y=\sin(x)$, the key points in one period $[0,2\pi]$ are $(0,0),(\frac{\pi}{2},1),(\pi,0),(\frac{3\pi}{2}, - 1),(2\pi,0)$. For $y = 3\sin\left(x+\frac{\pi}{6}\right)-2$, we substitute $x$ values. Let $x+\frac{\pi}{6}=0$, then $x=-\frac{\pi}{6}$ and $y=-2$. Let $x+\frac{\pi}{6}=\frac{\pi}{2}$, then $x=\frac{\pi}{3}$ and $y=-2 + 3=1$. Let $x+\frac{\pi}{6}=\pi$, then $x=\frac{5\pi}{6}$ and $y=-2$. Let $x+\frac{\pi}{6}=\frac{3\pi}{2}$, then $x=\frac{4\pi}{3}$ and $y=-2-3=-5$. Let $x+\frac{\pi}{6}=2\pi$, then $x=\frac{11\pi}{6}$ and $y=-2$. We can then plot these points and draw a smooth sinusoidal curve with amplitude 3, period $2\pi$, phase - shift $\frac{\pi}{6}$ units to the left and vertical shift 2 units down on the given grid.

Answer:

The graph of the function $y=-2 + 3\sin\left(x+\frac{\pi}{6}\right)$ is a sinusoidal curve with amplitude 3, period $2\pi$, phase - shift $\frac{\pi}{6}$ units to the left and vertical shift 2 units down. (The actual graph should be drawn on the provided grid using the key - points found above).