Determine the amplitude and period of the following function. Then, graph the function. \[ y=-7 \sin \frac{1}{2} x \] The amplitude is \( \square \) . (Simplify your answer.)

**Step 1. Identify the amplitude** The general sine function is given by \[ y = a \sin(bx) \] where the amplitude is the absolute value of \(a\). In this function, \(a = -7\), so the amplitude is \[ \left| -7 \right| = 7. \] **Step 2. Identify the period** The period of a sine function is determined by the coefficient \(b\) inside the sine. The formula for the period is: \[ \text{Period} = \frac{2\pi}{\left| b \right|}. \] Here, \(b = \frac{1}{2}\), so the period becomes: \[ \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi. \] **Step 3. Graphing the function** We are given the function: \[ y = -7 \sin \frac{1}{2}x. \] Key features for graphing: - **Amplitude:** \(7\). This means the sine wave oscillates between \(-7\) and \(7\). - **Period:** \(4\pi\). One complete cycle occurs as \(x\) runs from \(0\) to \(4\pi\). - **Reflection:** The negative sign indicates that the sine curve is reflected across the \(x\)-axis. **Plotting guide:** 1. **Start at the origin:** At \(x = 0\), \(\sin(0) = 0\), so \(y = 0\). 2. **First quarter period:** At \(x = \pi\), calculate the argument: \[ \frac{1}{2}\pi = \frac{\pi}{2}. \] Since \(\sin\left(\frac{\pi}{2}\right) = 1\), then \(y = -7(1) = -7\). 3. **Half period:** At \(x = 2\pi\), the argument is: \[ \frac{1}{2}(2\pi) = \pi. \] And \(\sin \pi = 0\), so \(y = 0\). 4. **Third quarter period:** At \(x = 3\pi\), the argument is: \[ \frac{1}{2}(3\pi) = \frac{3\pi}{2}. \] And \(\sin\left(\frac{3\pi}{2}\right) = -1\), so \(y = -7(-1) = 7\). 5. **Complete period:** At \(x = 4\pi\), the argument is: \[ \frac{1}{2}(4\pi) = 2\pi. \] And \(\sin(2\pi) = 0\), so \(y = 0\). Plot these key points \((0, 0)\), \((\pi, -7)\), \((2\pi, 0)\), \((3\pi, 7)\), and \((4\pi, 0)\) and smoothly connect them in a sinusoidal fashion, remembering that the curve is reflected due to the negative sign. **Final Answer:** - The amplitude is \(7\). - The period is \(4\pi\).