Questions: Graph one complete cycle of the following. Label the axes accurately. y = -2 + 4 cos(1/2 x - π/3)

Solution Steps

The function to be graphed is: $$y = -2 + 4 \cos \left(\frac{1}{2} x - \frac{\pi}{3}\right)$$

The cosine function completes one cycle over an interval of $2\pi$. Given the argument of the cosine function is $\frac{1}{2} x - \frac{\pi}{3}$, we set: $$\frac{1}{2} x - \frac{\pi}{3} = 0 \quad \text{to} \quad \frac{1}{2} x - \frac{\pi}{3} = 2\pi$$ Solving for $x$: $$\frac{1}{2} x - \frac{\pi}{3} = 0 \implies x = \frac{2\pi}{3}$$ $$\frac{1}{2} x - \frac{\pi}{3} = 2\pi \implies \frac{1}{2} x = 2\pi + \frac{\pi}{3} \implies x = 4\pi + \frac{2\pi}{3} = \frac{14\pi}{3}$$ Thus, the range for one complete cycle is: $$x \in \left[\frac{2\pi}{3}, \frac{14\pi}{3}\right]$$

The cosine function ranges from -1 to 1. Therefore: $$y = -2 + 4 \cos \left(\frac{1}{2} x - \frac{\pi}{3}\right)$$ $$y_{\text{min}} = -2 + 4(-1) = -6$$ $$y_{\text{max}} = -2 + 4(1) = 2$$ Thus, the y-axis range is: $$y \in [-6, 2]$$

Final Answer