For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -½ cos 3x

검증된 단계별 안내

Identify the general form of the cosine function: \(y = A \cos(Bx - C) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical translation.

Find the amplitude by taking the absolute value of the coefficient in front of the cosine function: \(|A| = |-\frac{1}{2}|\).

Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) inside the cosine function.

Determine the vertical translation \(D\) by identifying any constant added or subtracted outside the cosine function. In this case, there is none, so \(D = 0\).

Find the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\), where \(C\) is the horizontal shift inside the cosine argument. Since there is no \(C\) term, the phase shift is zero.

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주요 개념

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

Amplitude of a Trigonometric Function

Amplitude measures the maximum distance a trigonometric function's graph reaches from its midline. For functions like y = a cos bx, the amplitude is the absolute value of 'a'. It determines the height of the peaks and depths of the troughs in the wave.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function. For y = cos(bx), the period is calculated as (2π) divided by the absolute value of 'b'. It indicates how frequently the wave repeats over the x-axis.

Phase Shift and Vertical Translation

Phase shift refers to the horizontal shift of the graph, determined by any horizontal addition or subtraction inside the function's argument. Vertical translation is the upward or downward shift of the graph, indicated by any constant added or subtracted outside the function. In y = -½ cos 3x, there is no phase shift or vertical translation since no horizontal or vertical constants are added.

For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.y = -½ cos 3x