Fill in the blank(s) to correctly complete each sentence.
The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x unit(s) __ (up/down).

검증된 단계별 안내

Identify the base function and the transformation: The base function is \(y = 2 \cos x\), and the given function is \(y = -5 + 2 \cos x\).

Recognize that adding or subtracting a constant outside the cosine function results in a vertical shift of the graph.

Since the function is \(y = 2 \cos x - 5\), this means the graph of \(y = 2 \cos x\) is shifted vertically by 5 units.

Determine the direction of the shift: because the constant is \(-5\), the graph shifts 5 units down.

Therefore, the graph of \(y = -5 + 2 \cos x\) is obtained by shifting the graph of \(y = 2 \cos x\) 5 units down.

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주요 개념

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

Vertical Shifts in Trigonometric Graphs

A vertical shift moves the entire graph of a function up or down without changing its shape. For y = 2 cos x, adding or subtracting a constant outside the cosine function shifts the graph vertically by that constant amount.

Amplitude and Vertical Translation

The amplitude of y = 2 cos x is 2, which affects the height of peaks and troughs. Adding -5 shifts the graph down by 5 units, changing the midline from y=0 to y=-5, but the amplitude remains the same.

Function Transformation Notation

In the function y = a cos x + d, 'd' represents the vertical shift. A positive 'd' shifts the graph up, while a negative 'd' shifts it down. Recognizing this helps identify how the graph moves relative to the base function y = a cos x.

Fill in the blank(s) to correctly complete each sentence.The graph of y = -5 + 2 cos x is obtained by shifting the graph of y = 2 cos x ________ unit(s) __________ (up/down).