Fill in the blank(s) to correctly complete each sentence.
The graph of y = -2 + 3 cos (x - π/6) is obtained by shifting the graph of y = cos x horizontally unit(s) to the , (right/left) stretching it vertically by a factor of , and then shifting it vertically unit(s) ____. (up/down)

검증된 단계별 안내

Identify the horizontal shift from the function inside the cosine: the term \( (x - \frac{\pi}{6}) \) indicates a horizontal shift. Since it is \( x - \frac{\pi}{6} \), the graph shifts \( \frac{\pi}{6} \) units to the right.

Determine the vertical stretch by looking at the coefficient multiplying the cosine function. The coefficient is 3, which means the graph is stretched vertically by a factor of 3.

Identify the vertical shift by looking at the constant term outside the cosine function. The term \( -2 \) means the graph is shifted 2 units down.

Summarize the transformations: start with the basic graph of \( y = \cos x \), shift it horizontally \( \frac{\pi}{6} \) units to the right, stretch it vertically by a factor of 3, and then shift it vertically 2 units down.

Fill in the blanks accordingly: horizontally \( \frac{\pi}{6} \) unit(s) to the right, vertically stretched by a factor of 3, and shifted vertically 2 unit(s) down.

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주요 개념

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

Horizontal Phase Shift in Trigonometric Functions

The horizontal shift of a cosine graph is determined by the value inside the function's argument, such as (x - π/6). This represents a shift to the right by π/6 units because subtracting a positive value inside the function moves the graph rightward along the x-axis.

Vertical Stretching and Amplitude

The coefficient multiplying the cosine function affects its amplitude, which is the height from the midline to a peak. A factor of 3 stretches the graph vertically by 3 times, making peaks and troughs three times farther from the midline compared to the basic cosine graph.

Vertical Shift of Trigonometric Graphs

Adding or subtracting a constant outside the cosine function shifts the graph vertically. In y = -2 + 3 cos(x - π/6), the -2 shifts the entire graph down by 2 units, moving the midline and all points accordingly along the y-axis.