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

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Identify the standard form of the sine function, which is \( y = a \sin(bx + c) + d \). In this case, \( y = -2 \sin(2\pi x) \).

Determine the amplitude of the function. The amplitude is the absolute value of the coefficient \( a \). Here, \( a = -2 \), so the amplitude is \(|-2| = 2\).

Find the period of the function. The period of a sine function is given by \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). In this case, \( b = 2\pi \), so the period is \( \frac{2\pi}{2\pi} = 1 \).

Graph the function over a two-period interval. Since the period is 1, a two-period interval would be from \( x = 0 \) to \( x = 2 \).

Plot key points of the sine function within the interval, considering the amplitude and period. The key points for one period of \( y = \sin(x) \) are at \( x = 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 \). Adjust these points for the given function \( y = -2 \sin(2\pi x) \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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(2πx), the period can be calculated by dividing the standard period by the coefficient of x, which in this case is 2π, resulting in a period of 1.

Amplitude of a Trigonometric Function

The amplitude of a trigonometric function refers to the maximum distance the function reaches from its midline. In the function y = -2 sin(2πx), the amplitude is given by the absolute value of the coefficient in front of the sine function, which is 2. This means the graph will oscillate between -2 and 2, but since the sine function is negated, it will reflect the wave vertically.

Graphing Sine Functions

Graphing sine functions involves plotting the values of the function over a specified interval. For y = -2 sin(2πx), the graph will show a wave that oscillates between -2 and 0, with a period of 1. Understanding how to identify key points, such as the maximum, minimum, and intercepts, is essential for accurately representing the function over the given interval.

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