Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = sin 3x

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, \( a = 1 \), \( b = 3 \), \( c = 0 \), 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. The period of a sine function is given by \( \frac{2\pi}{b} \). Here, \( b = 3 \), so the period is \( \frac{2\pi}{3} \).
4. To graph the function over a two-period interval, calculate the interval length as \( 2 \times \frac{2\pi}{3} = \frac{4\pi}{3} \).
5. Plot the function \( y = \sin(3x) \) over the interval \( [0, \frac{4\pi}{3}] \), marking key points 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(3x), the period is calculated by dividing the standard period by the coefficient of x, resulting in a period of 2π/3 for this function.

Amplitude of a Trigonometric Function

The amplitude of a trigonometric function refers to the maximum distance the function reaches 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(3x), 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(3x), one would plot points for x values within a two-period interval, which is from 0 to 4π. Understanding the period and amplitude is crucial for accurately representing the function's behavior on the graph.