Graph each function over a two-period interval. Give the period and amplitude. See Examples 2–5.
y = 2 sin ¼ x

Verified step by step guidance

1. Identify the standard form of the sine function: \( y = a \sin(bx + c) + d \). In this case, \( a = 2 \), \( b = \frac{1}{4} \), \( c = 0 \), and \( d = 0 \).
2. Determine the amplitude of the function. The amplitude is the absolute value of \( a \), which is \( |2| = 2 \).
3. Calculate the period of the function. The period of a sine function is given by \( \frac{2\pi}{b} \). Substitute \( b = \frac{1}{4} \) to find the period: \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).
4. Graph the function over a two-period interval. Since one period is \( 8\pi \), a two-period interval is \( 16\pi \). Plot the function from \( x = 0 \) to \( x = 16\pi \).
5. Sketch the sine wave, noting that it oscillates between \( -2 \) and \( 2 \) (the amplitude) and completes one full cycle every \( 8\pi \) units along the x-axis.

Key Concepts

Period of a Trigonometric Function

The period of a trigonometric function is the length of one complete cycle of the wave. For the sine function, the standard period is 2π. However, when the function is modified, such as in y = 2 sin(¼ x), the period can be calculated using the formula Period = 2π / |b|, where b is the coefficient of x. In this case, the period is 2π / (1/4) = 8π.

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 = 2 sin(¼ x), the amplitude is 2, meaning the graph will oscillate between 2 and -2.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting the values of the function over a specified interval. For y = 2 sin(¼ x), one must consider the calculated period and amplitude to accurately represent the wave. The graph will show oscillations between the maximum and minimum values defined by the amplitude, repeating every 8π units along the x-axis.