Graph each function over a two-period interval.
y = sin (x + π/4)

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Identify the basic function and its transformation. The given function is \(y = \sin\left(x + \frac{\pi}{4}\right)\), which is a sine function shifted horizontally.

Recall that the sine function \(y = \sin x\) has a period of \(2\pi\). Since there is no coefficient multiplying \(x\) inside the sine, the period remains \(2\pi\).

Determine the interval for two full periods. Since one period is \(2\pi\), two periods correspond to an interval of length \(4\pi\). You can choose the interval for \(x\) as \([a, a + 4\pi]\) for some \(a\).

Because of the phase shift \(+ \frac{\pi}{4}\) inside the sine, the graph is shifted to the left by \(\frac{\pi}{4}\). To capture two full periods starting from the standard position, set the interval for \(x\) as \([-\frac{\pi}{4}, -\frac{\pi}{4} + 4\pi]\).

Plot key points within this interval by evaluating \(y = \sin\left(x + \frac{\pi}{4}\right)\) at multiples of \(\frac{\pi}{2}\) (e.g., \(x = -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}, \ldots\)) to capture maxima, minima, and zeros, then sketch the smooth sine curve through these points.

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주요 개념

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

Sine Function and Its Graph

The sine function, y = sin(x), is a periodic wave oscillating between -1 and 1 with a period of 2π. Its graph is smooth and continuous, starting at zero when x = 0, reaching a maximum at π/2, zero at π, minimum at 3π/2, and returning to zero at 2π. Understanding this base shape is essential for graphing transformations.

Phase Shift in Trigonometric Functions

A phase shift occurs when the input variable x is replaced by (x + c), shifting the graph horizontally. For y = sin(x + π/4), the graph shifts left by π/4 units. Recognizing how this affects the starting point and key features of the sine wave is crucial for accurate graphing.

Period of a Trigonometric Function

The period is the length of one complete cycle of the function. For sine, the period is 2π, meaning the function repeats every 2π units. Graphing over a two-period interval means plotting the function from 0 to 4π (or any interval of length 4π), ensuring two full cycles are displayed.

Graph each function over a two-period interval.y = sin (x + π/4)