1. **State the problem:** We need to find the equation of a sine function in the form $$y = A \sin(Bx + C) + D$$ that fits the given graph with a point at $(3.25, 3.42)$, maximum near $4$, minimum near $-2$, and period roughly $2\pi$. 2. **Identify amplitude $A$ and vertical shift $D$:** - Amplitude $A$ is half the distance between maximum and minimum: $$A = \frac{4 - (-2)}{2} = \frac{6}{2} = 3$$ - Vertical shift $D$ is the midpoint between maximum and minimum: $$D = \frac{4 + (-2)}{2} = \frac{2}{2} = 1$$ 3. **Determine period and $B$:** - The period $T$ is roughly $2\pi$. - The formula relating $B$ and period is: $$T = \frac{2\pi}{|B|}$$ - Since $T = 2\pi$, then: $$2\pi = \frac{2\pi}{|B|} \implies |B| = 1$$ 4. **Find phase shift $C$ using the point $(3.25, 3.42)$:** - Substitute $x=3.25$, $y=3.42$, $A=3$, $B=1$, $D=1$ into the equation: $$3.42 = 3 \sin(1 \cdot 3.25 + C) + 1$$ - Isolate sine term: $$3.42 - 1 = 3 \sin(3.25 + C)$$ $$2.42 = 3 \sin(3.25 + C)$$ - Divide both sides by 3: $$\frac{2.42}{3} = \sin(3.25 + C)$$ - Simplify fraction: $$\sin(3.25 + C) = 0.8067$$ 5. **Solve for $3.25 + C$:** - Use inverse sine: $$3.25 + C = \arcsin(0.8067)$$ - Calculate approximate value: $$\arcsin(0.8067) \approx 0.94$$ 6. **Find $C$:** $$C = 0.94 - 3.25 = -2.31$$ 7. **Write the final equation:** $$y = 3 \sin(x - 2.31) + 1$$ This equation fits the given graph with the specified amplitude, period, vertical shift, and passes through the point $(3.25, 3.42)$.