Sine Function 310Faa

1. **State the problem:** We are given the function $$y=2\sin x + 1$$ and want to understand its properties. 2. **Formula and explanation:** The function is a sinusoidal function of the form $$y = A\sin x + D$$ where: - $$A=2$$ is the amplitude, which controls the height of the wave from the center line. - $$D=1$$ is the vertical shift, moving the sine wave up by 1 unit. 3. **Key properties:** - The amplitude $$|A|=2$$ means the sine wave oscillates between $$1-2 = -1$$ and $$1+2=3$$. - The period of $$\sin x$$ is $$2\pi$$, so this function also has period $$2\pi$$. - The midline is at $$y=1$$. 4. **Intermediate work:** - Maximum value: $$\max y = 2 \times 1 + 1 = 3$$ - Minimum value: $$\min y = 2 \times (-1) + 1 = -1$$ 5. **Summary:** The graph of $$y=2\sin x + 1$$ is a sine wave with amplitude 2, shifted up by 1, oscillating between -1 and 3, with period $$2\pi$$.