1. **State the problem:** We are given the function $y = 2 \sin\left(\frac{\pi}{4}(x + 3)\right) + 1$ and need to analyze its properties. 2. **Formula and explanation:** The general form of a sinusoidal function is $y = A \sin(B(x - C)) + D$ where: - $A$ is the amplitude (height from the center line to peak), - $B$ affects the period (frequency), - $C$ is the horizontal phase shift, - $D$ is the vertical shift. 3. **Identify parameters:** - Amplitude $A = 2$ (wave oscillates 2 units above and below the midline), - Inside sine, $B = \frac{\pi}{4}$, so period $T = \frac{2\pi}{B} = \frac{2\pi}{\pi/4} = 8$, - Phase shift $C = -3$ (since $x + 3$ means shift left by 3), - Vertical shift $D = 1$ (wave shifted up by 1 unit). 4. **Interpretation:** - The wave oscillates between $1 - 2 = -1$ and $1 + 2 = 3$ vertically. - The wave completes one full cycle every 8 units along the x-axis. - The wave is shifted left by 3 units. 5. **Summary:** The function describes a sine wave with amplitude 2, period 8, phase shift left 3, and vertical shift up 1. **Final answer:** The function $y = 2 \sin\left(\frac{\pi}{4}(x + 3)\right) + 1$ has amplitude 2, period 8, phase shift $-3$, and vertical shift 1.