\( 1 \leftarrow \quad \begin{array}{l}\text { Determine the amplitude and period of the function. Graph } \\ \text { the function. } \\ y=-\frac{1}{2} \cos \frac{\pi}{2} x \\ \text { The amplitude is } \square . \\ \text { (Simplify your answer.) }\end{array} \)

**Step 1. Write the function in standard form** The function is given by \[ y=-\frac{1}{2}\cos\left(\frac{\pi}{2}x\right) \] The standard form for a cosine function is \[ y=A\cos(Bx) \] where \(A\) is the amplitude (with the amplitude being \(|A|\)) and \(B\) controls the period. **Step 2. Identify the amplitude** Here, \[ A=-\frac{1}{2} \] The amplitude is \[ \left|-\frac{1}{2}\right|=\frac{1}{2} \] **Step 3. Determine the period** For a cosine function, the period is calculated by the formula \[ \text{Period}=\frac{2\pi}{\left|B\right|} \] In our function, \[ B=\frac{\pi}{2} \] Thus, \[ \text{Period}=\frac{2\pi}{\frac{\pi}{2}}=2\pi\times\frac{2}{\pi}=4 \] **Step 4. Graph description** - **Amplitude:** The graph oscillates between \(\frac{1}{2}\) and \(-\frac{1}{2}\) in the vertical direction. - **Reflection:** The negative sign in \(A\) indicates the graph is reflected over the horizontal axis compared to the standard cosine graph. - **Period:** The entire pattern repeats every 4 units along the \(x\)-axis. - **Key Points:** - At \(x=0\): \[ y=-\frac{1}{2}\cos(0)=-\frac{1}{2}(1)=-\frac{1}{2} \] - At \(x=1\): \[ y=-\frac{1}{2}\cos\left(\frac{\pi}{2}\cdot1\right)=-\frac{1}{2}\cos\left(\frac{\pi}{2}\right)=-\frac{1}{2}\cdot0=0 \] - At \(x=2\): \[ y=-\frac{1}{2}\cos\left(\pi\right)=-\frac{1}{2}(-1)=\frac{1}{2} \] - At \(x=3\): \[ y=-\frac{1}{2}\cos\left(\frac{3\pi}{2}\right)=-\frac{1}{2}\cdot0=0 \] - At \(x=4\): \[ y=-\frac{1}{2}\cos\left(2\pi\right)=-\frac{1}{2}\cdot1=-\frac{1}{2} \] These points help sketch one complete cycle of the graph from \(x=0\) to \(x=4\). **Final Answers** - The amplitude is \( \frac{1}{2} \). - The period is \( 4 \).