Graph the functions in Exercises 13–22. What is the period of each function?

sin (x/2)

검증된 단계별 안내

Identify the function: The function given is \( \sin(\frac{x}{2}) \). This is a sine function with a modified argument.

Determine the period of the sine function: The standard period of \( \sin(x) \) is \( 2\pi \). For \( \sin(\frac{x}{2}) \), the period is affected by the coefficient of \( x \) inside the sine function.

Calculate the new period: The period of \( \sin(kx) \) is \( \frac{2\pi}{|k|} \). Here, \( k = \frac{1}{2} \), so the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).

Graph the function: Start by plotting the key points of the sine function over one period \([0, 4\pi]\). These points include the intercepts, maximum, and minimum values.

Label the graph: Clearly indicate the period on the x-axis, and mark the amplitude (which remains 1 for this function) on the y-axis. The graph should show the wave completing one full cycle over the interval \([0, 4\pi]\).

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주요 개념

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

Period of a Function

The period of a function is the length of the interval over which the function repeats itself. For trigonometric functions like sine and cosine, the period is a key characteristic that determines how often the wave pattern occurs. For example, the standard sine function, sin(x), has a period of 2π, meaning it repeats every 2π units along the x-axis.

Transformation of Functions

Transformations involve changes to the basic form of a function, such as stretching, compressing, or shifting. In the case of sin(x/2), the 'x/2' indicates a horizontal stretch, which affects the period. Specifically, dividing the input by a factor (like 2) increases the period, resulting in a new period of 4π for sin(x/2).

Graphing Trigonometric Functions

Graphing trigonometric functions requires understanding their key features, including amplitude, period, and phase shift. For sin(x/2), the graph will oscillate between -1 and 1, with a period of 4π, meaning it will complete one full cycle over this interval. Recognizing these features helps in accurately sketching the function and predicting its behavior.

Graph the functions in Exercises 13–22. What is the period of each function?sin (x/2)