1. The problem is to graph the function $$y = 3 \sin \left( \frac{1}{2} x + \frac{\pi}{3} \right) - 1$$ which is a sine wave with transformations. 2. The general sine function is $$y = A \sin(Bx + C) + D$$ where: - $A$ is the amplitude (height of the wave peaks), - $B$ affects the period (frequency), - $C$ is the phase shift (horizontal shift), - $D$ is the vertical shift. 3. For this function: - Amplitude $A = 3$ means the wave oscillates 3 units above and below the midline. - Frequency factor $B = \frac{1}{2}$ means the period is $$\frac{2\pi}{B} = \frac{2\pi}{\frac{1}{2}} = 4\pi$$. - Phase shift is $$-\frac{C}{B} = -\frac{\frac{\pi}{3}}{\frac{1}{2}} = -\frac{\pi}{3} \times 2 = -\frac{2\pi}{3}$$, so the graph shifts left by $\frac{2\pi}{3}$. - Vertical shift $D = -1$ moves the whole graph down by 1 unit. 4. The midline of the wave is at $y = -1$. 5. The sine wave starts at $x = -\frac{2\pi}{3}$ due to phase shift, completes one full cycle over $4\pi$ units. 6. The maximum value is $3 + (-1) = 2$ and the minimum value is $-3 + (-1) = -4$. 7. Plotting this function will show a sine wave oscillating between 2 and -4, stretched horizontally to have period $4\pi$, shifted left by $\frac{2\pi}{3}$, and shifted down by 1. Final function to graph: $$y = 3 \sin \left( \frac{1}{2} x + \frac{\pi}{3} \right) - 1$$