1. **State the problem:** We are given a sinusoidal function of the form $$y = a \sin\bigl(b(x - c)\bigr) + d$$ and a graph description. We want to identify the parameters $a$, $b$, $c$, and $d$ based on the graph. 2. **Recall the formula and parameters:** - $a$ is the amplitude (half the distance between maximum and minimum values). - $b$ affects the period of the sine wave, with period $$T = \frac{2\pi}{b}$$. - $c$ is the horizontal shift (phase shift). - $d$ is the vertical shift (midline of the wave). 3. **Analyze the graph:** - The maximum value is approximately 6. - The minimum value is approximately -2. - The vertical shift $d$ is the midpoint between max and min: $$d = \frac{6 + (-2)}{2} = \frac{4}{2} = 2$$ - The amplitude $a$ is half the distance between max and min: $$a = \frac{6 - (-2)}{2} = \frac{8}{2} = 4$$ 4. **Determine the period and $b$:** - The period $T$ is about $\pi$ (the wave repeats every $\pi$ units). - Using the period formula: $$T = \frac{2\pi}{b} \implies b = \frac{2\pi}{T} = \frac{2\pi}{\pi} = 2$$ 5. **Determine the horizontal shift $c$:** - The sine function normally starts at zero at $x=0$. - The graph shows a shift to the right by $c$ units. - Since the wave starts near $x = \frac{\pi}{2}$ (where sine normally peaks), and the sine function peaks at $\frac{\pi}{2}$ without shift, the phase shift $c$ is approximately 0. 6. **Final function:** $$y = 4 \sin\bigl(2(x - 0)\bigr) + 2 = 4 \sin(2x) + 2$$ This matches the graph's amplitude, period, and vertical shift. **Answer:** $$\boxed{y = 4 \sin(2x) + 2}$$