Problem 16 - Precalculus Essentials

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\sin \frac{4}{3} x$$

Step-by-Step Solution

Answer

The amplitude of the function \( -\sin \frac{4}{3} x \) is 1 and the period is \( \frac{3\pi}{2} \).

Step 1: Determining the Amplitude

The amplitude is the absolute value of the coefficient of the sin part. So in the given function \( -\sin \frac{4}{3} x \), the amplitude is \( |-1| = 1 \).

Step 2: Determining the Period

The period is obtained by dividing \(2\pi\) by the absolute value of the coefficient of the x. So in the given function \( -\sin \frac{4}{3} x \), the period is \( \frac {2\pi}{\left |\frac{4}{3} \right |} = \frac{2\pi \times 3}{4} = \frac{3\pi}{2} \).

Step 3: Graphing the Function

The graph of one period of the function will have the form of a sine wave with a height (peak to trough) of 1 (amplitude) and a length (left to right) of \( \frac{3\pi}{2} \) (period). It will be a downward wave (negative) because of the negative coefficient on the sin function.