1. **State the problem:** We need to write the equation of a sine or cosine function that matches the given graph. 2. **Identify key features from the graph:** - Maxima at $\left(\frac{\pi}{3},0\right)$ and $\left(\frac{5\pi}{3},0\right)$ - Minima at $\left(-\frac{\pi}{3},-6\right)$ and $\left(\pi,-6\right)$ - Midline at $y = -3$ - Amplitude $= 3$ - Period $= 2\pi$ 3. **Recall the general form of cosine function:** $$y = A \cos(B(x - C)) + D$$ where - $A$ is amplitude - $B = \frac{2\pi}{\text{period}}$ - $C$ is horizontal shift (phase shift) - $D$ is vertical shift (midline) 4. **Determine amplitude $A$ and vertical shift $D$:** - Amplitude $A = 3$ - Midline $D = -3$ 5. **Determine $B$ from the period:** $$B = \frac{2\pi}{2\pi} = 1$$ 6. **Find phase shift $C$ using maxima:** - Cosine normally has a maximum at $x=0$. - Given maximum at $x = \frac{\pi}{3}$, so phase shift $C = \frac{\pi}{3}$. 7. **Write the equation:** $$y = 3 \cos\left(x - \frac{\pi}{3}\right) - 3$$ 8. **Verify with minima:** - Minimum occurs at $x = \pi$. - Substitute $x=\pi$: $$y = 3 \cos\left(\pi - \frac{\pi}{3}\right) - 3 = 3 \cos\left(\frac{2\pi}{3}\right) - 3 = 3 \times \left(-\frac{1}{2}\right) - 3 = -\frac{3}{2} - 3 = -\frac{9}{2} = -4.5$$ - The graph shows minimum at $-6$, so amplitude might be larger. 9. **Recalculate amplitude using max and min:** - Max value $= 0$, min value $= -6$ - Midline $D = \frac{0 + (-6)}{2} = -3$ - Amplitude $A = 0 - (-3) = 3$ 10. **Check cosine value at minimum:** - Cosine at minimum should be $-1$: $$y_{min} = A \times (-1) + D = -3 + (-3) = -6$$ - Matches the graph. **Final answer:** $$y = 3 \cos\left(x - \frac{\pi}{3}\right) - 3$$