Sinusoidal Equation 8Ec920

1. **State the problem:** We need to find an equation of the form $y = a \sin bx$ or $y = a \cos bx$ that matches the given sinusoidal graph. 2. **Identify key features:** - Midline: $y = 0$ - Amplitude: $4$ - Period: $3\pi$ - Passes through origin increasing (this suggests a sine function since $\sin 0 = 0$ and increasing at $0$) 3. **Recall formulas:** - Amplitude $a$ is the coefficient in front of sine or cosine. - Period $T$ relates to $b$ by the formula $$T = \frac{2\pi}{b}$$ 4. **Calculate $b$ using the period:** $$3\pi = \frac{2\pi}{b} \implies b = \frac{2\pi}{3\pi} = \frac{2}{3}$$ 5. **Write the equation:** Since the graph passes through the origin increasing, use sine: $$y = 4 \sin \left( \frac{2}{3} x \right)$$ 6. **Verify maxima and minima:** - Maxima occur when $\sin \left( \frac{2}{3} x \right) = 1$. - Solve for $x$: $$\frac{2}{3} x = \frac{\pi}{2} + 2k\pi \implies x = \frac{3}{2} \left( \frac{\pi}{2} + 2k\pi \right) = \frac{3\pi}{4} + 3k\pi$$ - For $k=0$, $x=\frac{3\pi}{4}$ matches the given maxima. - Minima occur when $\sin \left( \frac{2}{3} x \right) = -1$. - Solve for $x$: $$\frac{2}{3} x = \frac{3\pi}{2} + 2k\pi \implies x = \frac{3}{2} \left( \frac{3\pi}{2} + 2k\pi \right) = \frac{9\pi}{4} + 3k\pi$$ - For $k=0$, $x=\frac{9\pi}{4}$ matches the given minima. **Final answer:** $$y = 4 \sin \left( \frac{2}{3} x \right)$$