1. **State the problem:** We are given the function $y=4\cos\left(2x+\frac{2\pi}{3}\right)-1$ and want to understand its properties. 2. **Formula and explanation:** The function is a cosine function with amplitude, phase shift, and vertical shift. The general form is $y=A\cos(Bx+C)+D$ where: - $A$ is the amplitude (height of peaks), - $B$ affects the period (length of one cycle), - $C$ is the phase shift (horizontal shift), - $D$ is the vertical shift. 3. **Identify parameters:** - Amplitude $A=4$ - Angular frequency $B=2$ - Phase shift $C=\frac{2\pi}{3}$ - Vertical shift $D=-1$ 4. **Period calculation:** The period $T$ is given by $T=\frac{2\pi}{|B|}=\frac{2\pi}{2}=\pi$ 5. **Phase shift calculation:** Phase shift $= -\frac{C}{B} = -\frac{\frac{2\pi}{3}}{2} = -\frac{\pi}{3}$ This means the graph shifts to the left by $\frac{\pi}{3}$ units. 6. **Vertical shift:** The graph is shifted down by 1 unit. 7. **Summary:** - Amplitude: 4 - Period: $\pi$ - Phase shift: left $\frac{\pi}{3}$ - Vertical shift: down 1 This fully describes the function's behavior. **Final answer:** The function $y=4\cos\left(2x+\frac{2\pi}{3}\right)-1$ has amplitude 4, period $\pi$, phase shift left $\frac{\pi}{3}$, and vertical shift down 1.