Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = sin ⅔ x

Verified step by step guidance

1. Identify the standard form of the sine function, which is \( y = a \sin(bx + c) + d \). In this case, \( a = 1 \), \( b = \frac{2}{3} \), \( c = 0 \), and \( d = 0 \).
2. Determine the amplitude of the function. The amplitude is the absolute value of \( a \), which is \( |1| = 1 \).
3. Calculate the period of the function using the formula \( \text{Period} = \frac{2\pi}{|b|} \). Substitute \( b = \frac{2}{3} \) to find the period.
4. Graph the function over a two-period interval. First, calculate the period from the previous step, then plot the sine wave starting from \( x = 0 \) to \( x = 2 \times \text{Period} \).
5. Label the key points on the graph, such as the maximum, minimum, and intercepts, based on the calculated period and amplitude.

Key Concepts

Period of a Trigonometric Function

The period of a trigonometric function is the length of one complete cycle of the function. For the sine function, the standard period is 2π. When the function is modified, such as in y = sin(⅔ x), the period is calculated by dividing the standard period by the coefficient of x, resulting in a period of 2π/(⅔) = 3π.

Amplitude of a Trigonometric Function

Amplitude refers to the maximum height of the wave from its midline. For the sine function, the amplitude is determined by the coefficient in front of the sine term. In the function y = sin(⅔ x), the amplitude is 1, indicating that the graph oscillates between 1 and -1.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the values of the function over a specified interval. For y = sin(⅔ x), one would plot points for x values within a two-period interval (0 to 6π) and connect them smoothly to illustrate the wave-like nature of the sine function, taking into account the calculated period and amplitude.