1. **State the problem:** Graph the function $$y = \frac{3}{2} \sin \left( 2 \left( x + \frac{\pi}{4} \right) \right)$$ over one period and find key coordinates. 2. **Recall the formula for sine function period:** The period of $$y = \sin(bx)$$ is $$\frac{2\pi}{|b|}$$. 3. **Calculate the period:** Here, $$b = 2$$, so the period is $$\frac{2\pi}{2} = \pi$$. 4. **Determine the interval for one period:** Since the function is shifted by $$-\frac{\pi}{4}$$ inside the sine, the one-period interval for $$x$$ is from $$-\frac{\pi}{4}$$ to $$-\frac{\pi}{4} + \pi = \frac{3\pi}{4}$$. 5. **Find key points:** The sine function has key points at $$0, \frac{\pi}{2}, \pi$$ in its argument. Set the inside of sine equal to these: $$2 \left( x + \frac{\pi}{4} \right) = t$$ where $$t = 0, \frac{\pi}{2}, \pi$$. Solve for $$x$$: - For $$t=0$$: $$2 \left( x + \frac{\pi}{4} \right) = 0 \Rightarrow x = -\frac{\pi}{4}$$ - For $$t=\frac{\pi}{2}$$: $$2 \left( x + \frac{\pi}{4} \right) = \frac{\pi}{2} \Rightarrow x + \frac{\pi}{4} = \frac{\pi}{4} \Rightarrow x=0$$ - For $$t=\pi$$: $$2 \left( x + \frac{\pi}{4} \right) = \pi \Rightarrow x + \frac{\pi}{4} = \frac{\pi}{2} \Rightarrow x = \frac{\pi}{4}$$ 6. **Calculate corresponding $$y$$ values:** - At $$x = -\frac{\pi}{4}$$: $$y = \frac{3}{2} \sin(0) = 0$$ - At $$x = 0$$: $$y = \frac{3}{2} \sin\left( 2 \times \frac{\pi}{4} \right) = \frac{3}{2} \sin\left( \frac{\pi}{2} \right) = \frac{3}{2} \times 1 = \frac{3}{2}$$ - At $$x = \frac{\pi}{4}$$: $$y = \frac{3}{2} \sin(\pi) = 0$$ 7. **Additional points:** - At $$x = \frac{\pi}{8}$$ (midpoint between 0 and $$\frac{\pi}{4}$$): $$y = \frac{3}{2} \sin\left( 2 \left( \frac{\pi}{8} + \frac{\pi}{4} \right) \right) = \frac{3}{2} \sin\left( 2 \times \frac{3\pi}{8} \right) = \frac{3}{2} \sin\left( \frac{3\pi}{4} \right) = \frac{3}{2} \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{4} \approx 1.06$$ - At $$x = -\frac{\pi}{8}$$ (midpoint between $$-\frac{\pi}{4}$$ and 0): $$y = \frac{3}{2} \sin\left( 2 \left( -\frac{\pi}{8} + \frac{\pi}{4} \right) \right) = \frac{3}{2} \sin\left( 2 \times \frac{\pi}{8} \right) = \frac{3}{2} \sin\left( \frac{\pi}{4} \right) = \frac{3}{2} \times \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{4} \approx 1.06$$ 8. **Summary of key points:** | x | y | |---|---| | $$-\frac{\pi}{4}$$ | 0 | | $$-\frac{\pi}{8}$$ | $$\approx 1.06$$ | | 0 | $$\frac{3}{2} = 1.5$$ | | $$\frac{\pi}{8}$$ | $$\approx 1.06$$ | | $$\frac{\pi}{4}$$ | 0 | 9. **Graph description:** The graph is a sine wave starting at 0 at $$x = -\frac{\pi}{4}$$, rising to a maximum of 1.5 at $$x=0$$, and returning to 0 at $$x=\frac{\pi}{4}$$, completing one period.