Determine the amplitude and period of the following function. Then, graph the function. \( y=-7 \boldsymbol{\operatorname { s i n }} 4 \pi x \) (Simplify your answer.) The period is \( \frac{1}{2} \). (Type an exact answer using \( \pi \) as needed. Use integers or fractions for any numbers in the expression.) \( \begin{array}{l}\text { Use the graphing tool to graph the function. } \\ \text { Click to } \\ \text { enlarge } \\ \text { graph }\end{array} \)

**Step 1. Identify the general form** The given function is \[ y = -7 \sin(4\pi x) \] which can be compared to the general sine function: \[ y = A \sin(Bx) \] **Step 2. Determine the amplitude** The amplitude is given by the absolute value of \( A \). Here, \[ A = -7 \quad \Longrightarrow \quad \text{amplitude} = | -7 | = 7. \] **Step 3. Determine the period** The period of a sine function is calculated using the formula: \[ \text{Period} = \frac{2\pi}{|B|} \] In our function, \[ B = 4\pi, \] so the period is: \[ \text{Period} = \frac{2\pi}{4\pi} = \frac{1}{2}. \] **Step 4. Graphing the function** The function \( y = -7 \sin(4\pi x) \) has the following characteristics: - **Amplitude:** \( 7 \) - **Period:** \( \frac{1}{2} \) - The negative sign indicates a reflection in the horizontal axis. To graph the function: - Plot the \( x \)-axis in increments relative to the period \( \frac{1}{2} \). - The sine curve normally starts at \( (0,0) \). Due to the negative amplitude, it initially decreases. - Mark key points within one period: starting at \( (0, 0) \), the minimum at \( \left(\frac{1}{8}, -7\right) \), the next zero crossing at \( \left(\frac{1}{4}, 0\right) \), the maximum at \( \left(\frac{3}{8}, 7\right) \), and returning to zero at \( \left(\frac{1}{2}, 0\right) \). - Repeat the pattern for additional periods. **Final Answer** - Amplitude: \( 7 \) - Period: \( \frac{1}{2} \)