91Ó°ÊÓ

The amplitude of a function is the coefficient of the trigonometric function. In this case, the amplitude is the coefficient of the sine function, which is -3. The amplitude is always a positive value, so we take the absolute value of -3, which is 3.

The period of a sine function is determined by the coefficient of the x term, inside the function. The general form of a sine function is \(y = A \sin(Bx)\), where A is the amplitude and B is the coefficient of x. The period T is given by the formula: \(T = \frac{2\pi}{B}\) In our given function, \(y = -3 \sin(-4x)\), B is equal to -4. The period T is then calculated as follows: \(T = \frac{2\pi}{-4} = -\frac{\pi}{2}\). Since the period is always positive, we take the absolute value of \(-\frac{\pi}{2}\), which is \(\frac{\pi}{2}\).

To sketch the graph over one period, follow these steps: 1. Label the x-axis with angles - this will help in plotting the sine function. 2. The amplitude of the function is 3, so the graph will oscillate between -3 and 3 on the y-axis. 3. The period is \(\frac{\pi}{2}\), which means the sine function will complete one cycle from 0 to \(\frac{\pi}{2}\). 4. Since there's a negative sign in front of the sine function, it is a reflection of the sine curve over the x-axis, and starts with a negative value at x = 0. 5. Considering the negative factor within sine function \(-4x\), the graph is reflected over the y-axis as well. Now, plot the graph using the following key points: - At x = 0, y = -3. - At x = \(\frac{\pi}{4}\), y = 0. - At x = \(\frac{\pi}{2}\), y = 3. By connecting these points, draw a smooth curve, which completes the graph of the function over one period.