Determine the amplitude and period of the function. Graph the function. \( y=-5 \cos \frac{2}{3} x \) The amplitude is \( \square \). (Simplify your answer.)

1. The given function is \[ y=-5 \cos\frac{2}{3}x. \] 2. **Amplitude:** The amplitude is the absolute value of the coefficient in front of the cosine function. Here, the coefficient is \(-5\), so \[ \text{Amplitude} = |{-5}| = 5. \] 3. **Period:** For a cosine function of the form \[ y = A \cos(Bx), \] the period \(P\) is given by \[ P=\frac{2\pi}{|B|}. \] In our function, \(B=\frac{2}{3}\), so \[ P=\frac{2\pi}{\frac{2}{3}} = 2\pi \times \frac{3}{2} = 3\pi. \] 4. **Graphing the Function:** - The basic cosine graph \(y=\cos x\) has been reflected over the \(x\)-axis because of the negative sign, and it is vertically stretched by a factor of 5. - The maximum value of \(-5 \cos\frac{2}{3}x\) is \(5\) and the minimum value is \(-5\). - One period of the function, spanning from \(0\) to \(3\pi\), shows the typical wave behavior with: - A starting point at \(x=0\) where \(y=-5 \cos(0)= -5\). - A mid-point at \(x=\frac{3\pi}{2}\) where the cosine function reaches its opposite from the start. - And returning to the same value at \(x=3\pi\). The amplitude is \(5\).