Sinusoidal Function 840Db4

1. **State the problem:** We are given the function $$y=1-3\sin2\left(x+\frac{\pi}{3}\right)$$ and we want to analyze it. 2. **Formula and explanation:** This is a sinusoidal function of the form $$y = A - B \sin(C(x+D))$$ where: - $$A=1$$ is the vertical shift, - $$B=3$$ is the amplitude, - $$C=2$$ affects the period, - $$D=\frac{\pi}{3}$$ is the horizontal phase shift. 3. **Period calculation:** The period $$T$$ of $$\sin(Cx)$$ is $$\frac{2\pi}{C}$$. $$T = \frac{2\pi}{2} = \pi$$ 4. **Amplitude:** The amplitude is the absolute value of the coefficient in front of sine: $$|B| = 3$$ 5. **Phase shift:** The phase shift is given by $$-D$$: $$-\frac{\pi}{3}$$ (shift to the left by $$\frac{\pi}{3}$$). 6. **Vertical shift:** The graph is shifted up by $$1$$ unit. 7. **Summary:** - Amplitude: $$3$$ - Period: $$\pi$$ - Phase shift: left $$\frac{\pi}{3}$$ - Vertical shift: up $$1$$ This function oscillates between $$1-3= -2$$ and $$1+3=4$$ with period $$\pi$$ and is shifted left by $$\frac{\pi}{3}$$.