1. **Problem Statement:** Sketch one full period of the function $y = 2 \sin(x) + 3$. 2. **Formula and Important Rules:** The general sine function is $y = A \sin(Bx - C) + D$ where: - $A$ is the amplitude (height of the wave peaks from the center line). - $B$ affects the period (length of one full cycle), period $= \frac{2\pi}{|B|}$. - $C$ is the horizontal phase shift. - $D$ is the vertical shift. 3. **Identify parameters for $y = 2 \sin(x) + 3$:** - Amplitude $A = 2$. - Frequency factor $B = 1$, so period $= \frac{2\pi}{1} = 2\pi$. - Phase shift $C = 0$. - Vertical shift $D = 3$. 4. **Sketching one full period:** - The sine wave oscillates between $3 - 2 = 1$ and $3 + 2 = 5$ vertically. - One full period spans from $x = 0$ to $x = 2\pi$. - Key points: - At $x=0$, $y = 2 \sin(0) + 3 = 3$ (midline). - At $x=\frac{\pi}{2}$, $y = 2 \sin(\frac{\pi}{2}) + 3 = 2(1) + 3 = 5$ (maximum). - At $x=\pi$, $y = 2 \sin(\pi) + 3 = 3$ (midline). - At $x=\frac{3\pi}{2}$, $y = 2 \sin(\frac{3\pi}{2}) + 3 = 2(-1) + 3 = 1$ (minimum). - At $x=2\pi$, $y = 2 \sin(2\pi) + 3 = 3$ (midline). 5. **Explanation:** The sine wave is shifted up by 3 units, so the midline is at $y=3$ instead of $y=0$. The amplitude of 2 means the wave reaches 2 units above and below this midline. 6. **Final answer:** The graph of $y = 2 \sin(x) + 3$ is a sine wave with amplitude 2, period $2\pi$, and vertical shift 3, oscillating between 1 and 5 over one full period from $0$ to $2\pi$.