Fill in the blank(s) to correctly complete each sentence.
The graph of y = -3 sin x is obtained by stretching the graph of y = sin x by a factor of and reflecting across the -axis.

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Identify the general form of the sine function transformation: \(y = A \sin x\), where \(A\) represents the amplitude, which affects the vertical stretch or compression of the graph.

Compare the given function \(y = -3 \sin x\) with the basic sine function \(y = \sin x\). Notice that the amplitude changes from 1 to 3, indicating a vertical stretch by a factor of 3.

Observe the negative sign in front of the amplitude, which means the graph is reflected across the x-axis.

Summarize the transformation: the graph is stretched vertically by a factor of 3 and reflected across the x-axis.

Fill in the blanks accordingly: The graph of \(y = -3 \sin x\) is obtained by stretching the graph of \(y = \sin x\) by a factor of 3 and reflecting across the x-axis.

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주요 개념

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

Amplitude and Vertical Stretching

The amplitude of a sine function is the absolute value of the coefficient before sin x, which determines the vertical stretch or compression. For y = -3 sin x, the amplitude is 3, meaning the graph is stretched vertically by a factor of 3 compared to y = sin x.

Reflection Across the x-axis

A negative coefficient before the sine function indicates a reflection of the graph across the x-axis. This flips the graph upside down, changing the direction of the peaks and troughs while preserving the shape.

Basic Sine Function Graph

The graph of y = sin x is a periodic wave oscillating between -1 and 1 with a period of 2π. Understanding this baseline graph is essential to recognize how transformations like stretching and reflection affect its shape.

Fill in the blank(s) to correctly complete each sentence.The graph of y = -3 sin x is obtained by stretching the graph of y = sin x by a factor of ________ and reflecting across the ________-axis.