1. **State the problem:** We need to analyze and sketch the graph of the function $$y = -2 \cos\left(0.25x - \frac{\pi}{8}\right) + 3$$ by finding its amplitude, axis of the curve, period, phase shift, and then determine points on the graph. 2. **Recall the general form and formulas:** The general cosine function is $$y = A \cos(Bx - C) + D$$ where: - Amplitude = $$|A|$$ - Axis of the curve (midline) = $$y = D$$ - Period = $$\frac{2\pi}{|B|}$$ - Phase shift = $$\frac{C}{B}$$ 3. **Identify parameters from the given function:** - $$A = -2$$ - $$B = 0.25$$ - $$C = \frac{\pi}{8}$$ - $$D = 3$$ 4. **Calculate amplitude:** $$\text{Amplitude} = |A| = |-2| = 2$$ 5. **Equation of the axis of the curve:** $$y = D = 3$$ 6. **Calculate the period:** $$\text{Period} = \frac{2\pi}{|B|} = \frac{2\pi}{0.25} = 8\pi$$ 7. **Calculate the phase shift:** $$\text{Phase shift} = \frac{C}{B} = \frac{\frac{\pi}{8}}{0.25} = \frac{\pi}{8} \times 4 = \frac{\pi}{2}$$ This means the graph shifts to the right by $$\frac{\pi}{2}$$ units relative to $$y = \cos(x)$$. 8. **Find key points on one period:** The cosine function starts at its maximum at phase shift, then goes through zero, minimum, zero, and back to maximum over one period. - Start point (maximum): $$x = \frac{\pi}{2}$$ $$y = -2 \cos(0.25 \times \frac{\pi}{2} - \frac{\pi}{8}) + 3 = -2 \cos\left(\frac{\pi}{8} - \frac{\pi}{8}\right) + 3 = -2 \cos(0) + 3 = -2(1) + 3 = 1$$ - Quarter period later: $$x = \frac{\pi}{2} + \frac{8\pi}{4} = \frac{\pi}{2} + 2\pi = \frac{\pi}{2} + 6.2832 \approx 7.854$$ $$y = -2 \cos(0.25 \times 7.854 - \frac{\pi}{8}) + 3 = -2 \cos(1.9635 - 0.3927) + 3 = -2 \cos(1.5708) + 3 = -2(0) + 3 = 3$$ - Half period later (minimum): $$x = \frac{\pi}{2} + 4\pi = \frac{\pi}{2} + 12.5664 \approx 14.137$$ $$y = -2 \cos(0.25 \times 14.137 - \frac{\pi}{8}) + 3 = -2 \cos(3.534 - 0.393) + 3 = -2 \cos(3.1416) + 3 = -2(-1) + 3 = 5$$ - Three quarters period later: $$x = \frac{\pi}{2} + 6\pi = \frac{\pi}{2} + 18.8496 \approx 20.420$$ $$y = -2 \cos(0.25 \times 20.420 - \frac{\pi}{8}) + 3 = -2 \cos(5.105 - 0.393) + 3 = -2 \cos(4.712) + 3 = -2(0) + 3 = 3$$ - Full period later (back to max): $$x = \frac{\pi}{2} + 8\pi = \frac{\pi}{2} + 25.1327 \approx 25.704$$ $$y = -2 \cos(0.25 \times 25.704 - \frac{\pi}{8}) + 3 = -2 \cos(6.426 - 0.393) + 3 = -2 \cos(6.033) + 3 \approx -2(0.994) + 3 = 1.012$$ 9. **Summary of points:** | x | y | |---|---| | $$\frac{\pi}{2} \approx 1.571$$ | 1 | | 7.854 | 3 | | 14.137 | 5 | | 20.420 | 3 | | 25.704 | 1.012 | These points can be plotted to sketch the graph. --- **Final answers:** - Amplitude = 2 - Axis of the curve: $$y = 3$$ - Period = $$8\pi$$ - Phase shift = $$\frac{\pi}{2}$$ to the right