1. **State the problem:** We have a curve defined by the equation $$y = a + b \sin(cx)$$ with amplitude 4 and period $$\frac{\pi}{3}$$. The curve passes through the point $$\left(\frac{\pi}{12}, 2\right)$$. We need to find the constants $$a$$, $$b$$, and $$c$$. 2. **Recall formulas and rules:** - Amplitude of $$y = a + b \sin(cx)$$ is $$|b|$$. - Period of $$y = a + b \sin(cx)$$ is $$\frac{2\pi}{c}$$. 3. **Use amplitude to find $$b$$:** Given amplitude = 4, so $$|b| = 4 \Rightarrow b = 4$$ (assuming positive for sine amplitude). 4. **Use period to find $$c$$:** Given period $$= \frac{\pi}{3}$$, so $$\frac{2\pi}{c} = \frac{\pi}{3} \Rightarrow c = \frac{2\pi}{\frac{\pi}{3}} = 2\pi \times \frac{3}{\pi} = 6$$. 5. **Use the point $$\left(\frac{\pi}{12}, 2\right)$$ to find $$a$$:** Substitute $$x = \frac{\pi}{12}$$ and $$y = 2$$ into the equation: $$2 = a + 4 \sin\left(6 \times \frac{\pi}{12}\right) = a + 4 \sin\left(\frac{6\pi}{12}\right) = a + 4 \sin\left(\frac{\pi}{2}\right)$$ Since $$\sin\left(\frac{\pi}{2}\right) = 1$$, $$2 = a + 4 \times 1 = a + 4$$ Subtract 4 from both sides: $$2 - 4 = a \Rightarrow a = -2$$ 6. **Final values:** $$a = -2, b = 4, c = 6$$ --- **Summary:** The curve is $$y = -2 + 4 \sin(6x)$$.