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Ch. 4 - Graphs of the Circular Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 4 - Graphs of the Circular FunctionsProblema 41
Capitolo 5, Problema 41

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

Guida verificata passo dopo passo
1
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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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.
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Percorso guidato
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Graph of Sine and Cosine Function

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.
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Percorso guidato
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Phase Shifts

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.
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Percorso guidato
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Period of Sine and Cosine Functions
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Graph each function over a two-period interval.

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y = -½ cos (πx - π)

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