Fill in the blank(s) to correctly complete each sentence.
The graph of y = 6 + 3 sin x is obtained by shifting the graph of y = 3 sin x unit(s) __ (up/down).

검증된 단계별 안내

Identify the base function and the transformation applied. The base function here is \(y = 3 \sin x\), and the transformed function is \(y = 6 + 3 \sin x\).

Recognize that adding a constant outside the sine function results in a vertical shift of the graph.

The constant added is 6, so the graph of \(y = 3 \sin x\) is shifted vertically by 6 units.

Since the constant is positive, the shift is upwards.

Therefore, the graph of \(y = 6 + 3 \sin x\) is obtained by shifting the graph of \(y = 3 \sin x\) 6 units up.

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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 a function y = f(x) + k, the graph shifts vertically by k units; if k is positive, the shift is upward, and if negative, downward.

Amplitude of a Sine Function

The amplitude of y = a sin x is the absolute value of a, representing the maximum distance from the midline to the peak or trough. In y = 3 sin x, the amplitude is 3, which remains unchanged by vertical shifts.

Midline of a Trigonometric Function

The midline is the horizontal line around which the sine or cosine graph oscillates. For y = 3 sin x, the midline is y = 0; adding 6 shifts the midline to y = 6, indicating a vertical shift upward by 6 units.

Fill in the blank(s) to correctly complete each sentence.The graph of y = 6 + 3 sin x is obtained by shifting the graph of y = 3 sin x ________ unit(s) __________ (up/down).