Fill in the blank(s) to correctly complete each sentence.
The graph of y = 3 + 5 cos (x + π/5) 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 function is y = 3 + 5 \(\cos\)(x + \(\frac{\pi}{5}\)). The horizontal shift is determined by the phase shift, which is found by setting the inside of the cosine to zero: x + \(\frac{\pi}{5}\) = 0, so x = -\(\frac{\pi}{5}\). This means the graph shifts horizontally by \(\frac{\pi}{5}\) units to the left.

Determine the vertical stretch by looking at the coefficient in front of the cosine function. The coefficient 5 means the graph is stretched vertically by a factor of 5 compared to the basic cosine graph y = \(\cos\) x.

Identify the vertical shift by looking at the constant added outside the cosine function. The +3 means the entire graph is shifted vertically 3 units up.

Summarize the transformations: horizontal shift of \(\frac{\pi}{5}\) units to the left, vertical stretch by a factor of 5, and vertical shift 3 units up.

Fill in the blanks accordingly: The graph is shifted horizontally \(\frac{\pi}{5}\) unit(s) to the left, stretched vertically by a factor of 5, and shifted vertically 3 unit(s) up.

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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 phase shift inside the function's argument. For y = cos(x + c), the graph shifts horizontally by |c| units to the left if c is positive, and to the right if c is negative. This shift moves the entire wave along the x-axis without changing its shape.

Vertical Stretching and Amplitude

The coefficient multiplying the cosine function affects its amplitude, which is the height from the midline to a peak. In y = a cos x, the amplitude is |a|. A value greater than 1 stretches the graph vertically, making peaks and troughs more pronounced, while a value between 0 and 1 compresses it.

Vertical Shift of Trigonometric Graphs

Adding or subtracting a constant outside the cosine function shifts the graph vertically. For y = cos x + d, the entire graph moves up by d units if d is positive, and down if d is negative. This changes the midline of the wave without affecting its amplitude or period.