Cosine Wave Shift Cc5Bb6

1. **State the problem:** We are given the function $$y = -1 + \frac{3}{2} \cos(2x - \pi)$$ and asked to understand its graph shape, including midline, amplitude, period, phase shift, and maxima positions.\n\n2. **Identify the midline:** The midline is the vertical shift of the cosine function, which is the constant term outside the cosine. Here, it is $$-1$$. So, the midline is $$y = -1$$.\n\n3. **Find the amplitude:** The amplitude is the coefficient in front of the cosine function, which is $$\frac{3}{2}$$. This means the wave oscillates $$\frac{3}{2}$$ units above and below the midline.\n\n4. **Calculate the period:** The period of a cosine function $$\cos(bx)$$ is given by $$\frac{2\pi}{|b|}$$. Here, $$b = 2$$, so the period is $$\frac{2\pi}{2} = \pi$$.\n\n5. **Determine the phase shift:** The phase shift is found by solving $$2x - \pi = 0$$ for $$x$$, which gives $$x = \frac{\pi}{2}$$. This means the graph is shifted to the right by $$\frac{\pi}{2}$$.\n\n6. **Locate maxima:** The cosine function has maxima where its argument equals multiples of $$2\pi$$. The argument is $$2x - \pi$$. Setting $$2x - \pi = 2k\pi$$ for integers $$k$$, we get $$x = \frac{\pi}{2} + k\pi$$. For $$k=0$$, maximum at $$x = \frac{\pi}{2}$$ (top-left or center depending on view). For $$k=1$$, maximum near $$x = \frac{\pi}{2} + \pi = \frac{3\pi}{2}$$, which is near $$\pi$$ as stated.\n\n7. **Summary:** The graph is a cosine wave with midline $$y = -1$$, amplitude $$\frac{3}{2}$$, period $$\pi$$, shifted right by $$\frac{\pi}{2}$$, with maxima at $$x = \frac{\pi}{2} + k\pi$$.