1. The problem involves analyzing the given trigonometric functions to identify their midline, amplitude, period, and phase shift. 2. Recall the general form of sine and cosine functions: $$f(x) = A \sin(B(x - C)) + D \quad \text{or} \quad f(x) = A \cos(B(x - C)) + D$$ where: - $A$ is the amplitude (height from midline to peak), - $\frac{2\pi}{B}$ is the period, - $C$ is the phase shift, - $D$ is the vertical shift or midline. 3. Analyze the first function: $$f(x) = -4 \sin\left(\frac{2}{3}(x - \frac{\pi}{2})\right)$$ - Amplitude $= |A| = 4$ - Midline $D = 0$ - Period $= \frac{2\pi}{B} = \frac{2\pi}{\frac{2}{3}} = 3\pi$ - Phase shift $= C = \frac{\pi}{2}$ Note: The problem states phase shift as $\frac{3\pi}{2}$ which is incorrect; it should be $\frac{\pi}{2}$. 4. Analyze the second function: $$f(x) = -3 \cos\left(6\pi \left(x + \frac{5}{2}\right)\right) + 4$$ - Amplitude $= 3$ - Midline $= 4$ - Period $= \frac{2\pi}{6\pi} = \frac{1}{3}$ - Phase shift $= -\frac{5}{2}$ (since $x + \frac{5}{2} = x - (-\frac{5}{2})$) 5. Analyze the third function: $$f(x) = 4 \cos\left(\frac{3}{2}x - \frac{15\pi}{4}\right) - 4$$ Rewrite inside cosine as: $$\frac{3}{2} \left(x - \frac{15\pi}{4} \times \frac{2}{3}\right) = \frac{3}{2} \left(x - \frac{5\pi}{2}\right)$$ - Amplitude $= 4$ - Midline $= -4$ - Period $= \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3}$ - Phase shift $= \frac{5\pi}{2}$ 6. Analyze the fourth function: $$f(x) = 1 + 2 \sin\left(\frac{\pi}{4}x - \frac{3\pi}{16}\right)$$ Rewrite inside sine as: $$\frac{\pi}{4} \left(x - \frac{3\pi}{16} \times \frac{4}{\pi}\right) = \frac{\pi}{4} \left(x - \frac{3}{4}\right)$$ - Amplitude $= 2$ - Midline $= 1$ - Period $= \frac{2\pi}{\frac{\pi}{4}} = 8$ - Phase shift $= \frac{3}{4}$ 7. Summary: | Function | Midline | Amplitude | Period | Phase Shift | |---|---|---|---|---| | 1 | $0$ | $4$ | $3\pi$ | $\frac{\pi}{2}$ | | 2 | $4$ | $3$ | $\frac{1}{3}$ | $-\frac{5}{2}$ | | 3 | $-4$ | $4$ | $\frac{4\pi}{3}$ | $\frac{5\pi}{2}$ | | 4 | $1$ | $2$ | $8$ | $\frac{3}{4}$ | The original table had some missing or incorrect entries; this completes and corrects them.