Fill in the blank(s) to correctly complete each sentence.
The graph of y = 4 sin x is obtained by stretching the graph of y = sin x vertically by a factor of ________.

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Identify the base function and the transformed function. Here, the base function is \(y = \sin x\) and the transformed function is \(y = 4 \sin x\).

Recall that the coefficient in front of the sine function affects the amplitude, which is the vertical stretch or compression of the graph.

The amplitude of \(y = \sin x\) is 1, since the sine function oscillates between -1 and 1.

The amplitude of \(y = 4 \sin x\) is 4, meaning the graph is stretched vertically by a factor equal to the amplitude.

Therefore, the vertical stretch factor is the ratio of the new amplitude to the original amplitude, which is \(4\).

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주요 개념

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

Amplitude of a Sine Function

The amplitude of a sine function y = a sin x is the absolute value of the coefficient 'a'. It represents the maximum vertical distance from the midline (usually the x-axis) to the peak of the wave, indicating how tall or stretched the graph is.

Vertical Stretching of Graphs

Vertical stretching occurs when the output values of a function are multiplied by a factor greater than 1, increasing the distance from the x-axis. For y = a sin x, if |a| > 1, the graph is stretched vertically by a factor of |a| compared to y = sin x.

Basic Sine Function Graph

The basic sine function y = sin x has an amplitude of 1 and a period of 2π. Understanding its shape and properties is essential to recognize how changes in coefficients affect the graph, such as vertical stretches or compressions.

Fill in the blank(s) to correctly complete each sentence.The graph of y = 4 sin x is obtained by stretching the graph of y = sin x vertically by a factor of ________.