1. **State the problem:** We need to analyze and understand the function $$f(x) = 2 \sin\left(\frac{\pi}{4} x\right) + 3$$ and sketch its graph. 2. **Formula and important rules:** The general sine function is $$y = A \sin(Bx + C) + D$$ where: - $A$ is the amplitude (height from the midline to peak), - $B$ affects the period (length of one cycle), - $C$ is the phase shift (horizontal shift), - $D$ is the vertical shift (midline). The period is calculated by $$\text{Period} = \frac{2\pi}{|B|}$$. 3. **Identify parameters:** - Amplitude $A = 2$ (the sine wave oscillates 2 units above and below the midline). - $B = \frac{\pi}{4}$. - Vertical shift $D = 3$ (the midline is at $y=3$). - No phase shift ($C=0$). 4. **Calculate the period:** $$\text{Period} = \frac{2\pi}{\frac{\pi}{4}} = 2\pi \times \frac{4}{\pi} = 8$$ 5. **Interpretation:** - The sine wave completes one full cycle every 8 units along the x-axis. - The wave oscillates between $3 - 2 = 1$ and $3 + 2 = 5$ on the y-axis. 6. **Summary:** - Midline: $y=3$ - Amplitude: 2 - Period: 8 - No horizontal shift This means the graph oscillates smoothly between 1 and 5, repeating every 8 units along the x-axis, centered vertically at 3.