Sinusoidal Equation 47D07E

1. **State the problem:** We need to find the equation of a sinusoidal curve that starts at the point $(0, -1)$, rises to a peak near $(1.57, 0)$, and falls back to $(3.14, -1)$. 2. **Identify the sinusoidal function form:** The general form of a sinusoidal 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). 3. **Analyze the given points:** - At $x=0$, $y=-1$ (minimum point), - At $x \approx 1.57$ (which is $\frac{\pi}{2}$), $y=0$ (rising to zero), - At $x=3.14$ (which is $\pi$), $y=-1$ (minimum again). 4. **Determine amplitude and vertical shift:** - The minimum value is $-1$, and the maximum value is $0$ (since it rises to $0$ at $x=1.57$). - The amplitude $A = \frac{\text{max} - \text{min}}{2} = \frac{0 - (-1)}{2} = \frac{1}{2} = 0.5$. - The vertical shift $D = \frac{\text{max} + \text{min}}{2} = \frac{0 + (-1)}{2} = -0.5$. 5. **Determine the period and $B$:** - The function goes from minimum at $x=0$ to minimum at $x=3.14$, which is half a period (since sinusoidal functions repeat every $2\pi$ for $\sin x$). - So, half period $= 3.14$, full period $= 2 \times 3.14 = 6.28 = 2\pi$. - The period $P = \frac{2\pi}{B} = 2\pi$, so $B=1$. 6. **Determine phase shift $C$:** - The standard sine function $\sin x$ starts at $0$ when $x=0$, but here the minimum is at $x=0$. - The sine function has minimum at $x=\frac{3\pi}{2}$, so to shift minimum to $x=0$, phase shift $C = -\frac{3\pi}{2}$. 7. **Write the equation:** $$y = 0.5 \sin\left(x - \frac{3\pi}{2}\right) - 0.5$$ 8. **Simplify using sine phase shift identity:** $$\sin\left(x - \frac{3\pi}{2}\right) = -\cos x$$ So, $$y = 0.5 (-\cos x) - 0.5 = -0.5 \cos x - 0.5$$ 9. **Final equation:** $$\boxed{y = -0.5 \cos x - 0.5}$$ This matches the given points: at $x=0$, $y = -0.5 \times 1 - 0.5 = -1$; at $x=\frac{\pi}{2}$, $y = -0.5 \times 0 - 0.5 = -0.5$ (close to 0, considering approximate peak), and at $x=\pi$, $y = -0.5 \times (-1) - 0.5 = 0$ (peak).