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, and phase shift relative to $$y = \cos(x)$$. 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 (vertical shift) = $$y = D$$ - Period = $$\frac{2\pi}{|B|}$$ - Phase shift = $$\frac{C}{B}$$ (shift to the right if positive, left if negative) 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. **Determine 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 $$\frac{\pi}{2}$$ units to the right relative to $$y = \cos(x)$$. 8. **Interpret the negative amplitude:** The negative sign in $$A$$ reflects the graph over the axis of the curve (vertical reflection). **Final summary:** - Amplitude: 2 - Axis of the curve: $$y = 3$$ - Period: $$8\pi$$ - Phase shift: $$\frac{\pi}{2}$$ units to the right - Reflection about the axis due to negative amplitude This information allows you to sketch the graph accurately.