1. **State the problem:** We are given a sine function in the form $$y = a \sin(b(x - c)) + d$$ and a graph with specific points. We need to find the value of $$a + b + c + d$$ to the nearest tenth. 2. **Identify parameters from the graph:** - The wave crosses the y=0 line at $$x = -3\pi, -\pi, \pi, 3\pi$$. - Maximum points at $$(-2\pi, 4)$$ and $$(2\pi, 4)$$. - Minimum points at $$(0, 0)$$ and $$(4\pi, 0)$$. 3. **Find vertical shift $$d$$:** The midline is halfway between max and min y-values. Max y = 4, Min y = 0 $$d = \frac{4 + 0}{2} = 2$$ 4. **Find amplitude $$a$$:** Amplitude is half the distance between max and min. $$a = \frac{4 - 0}{2} = 2$$ 5. **Find period and $$b$$:** The sine function crosses zero at $$x = -3\pi, -\pi, \pi, 3\pi$$, which are zeros spaced by $$2\pi$$. Distance between zeros is $$2\pi$$, so half period is $$2\pi$$. Full period $$T = 4\pi$$. Period formula: $$T = \frac{2\pi}{b} \Rightarrow b = \frac{2\pi}{T} = \frac{2\pi}{4\pi} = \frac{1}{2}$$ 6. **Find phase shift $$c$$:** The sine function normally crosses zero at $$x=0$$, but here it crosses zero at $$x=\pi$$ (one of the zeros). Using zero at $$x=\pi$$: $$b(x - c) = 0 \Rightarrow x - c = 0 \Rightarrow c = x = \pi$$ 7. **Calculate $$a + b + c + d$$:** $$a + b + c + d = 2 + \frac{1}{2} + \pi + 2 = 4.5 + \pi \approx 4.5 + 3.1416 = 7.6416$$ Rounded to nearest tenth: $$7.6$$ **Final answer:** $$7.6$$