Ch. 4 - Graphs of the Circular Functions

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 4 - Graphs of the Circular Functions

Problema 57

Capitolo 5, Problema 57

Graph each function over a two-period interval.
y = -2 + (1/2) sin 3x

Guida verificata passo dopo passo

Identify the given function: \(y = -2 + \frac{1}{2} \sin(3x)\).

Determine the period of the sine function. Recall that the period of \(\sin(bx)\) is \(\frac{2\pi}{b}\). Here, \(b = 3\), so the period is \(\frac{2\pi}{3}\).

Since the problem asks to graph over a two-period interval, calculate the interval length as \(2 \times \frac{2\pi}{3} = \frac{4\pi}{3}\).

Set up the x-axis interval for the graph, for example from \(x = 0\) to \(x = \frac{4\pi}{3}\), or any other interval of length \(\frac{4\pi}{3}\).

Plot key points by evaluating \(y\) at important values of \(x\) within the interval, such as at multiples of \(\frac{\pi}{6}\) or \(\frac{\pi}{3}\), considering the amplitude \(\frac{1}{2}\) and vertical shift \(-2\).

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Sine Function and Its Properties

The sine function, sin(x), is a periodic trigonometric function with a fundamental period of 2π. It oscillates between -1 and 1, producing a smooth wave. Understanding its shape and behavior is essential for graphing transformations and shifts.

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Graph of Sine and Cosine Function

Amplitude, Frequency, and Vertical Shift

Amplitude determines the height of the wave from its midline, frequency affects how many cycles occur in a given interval, and vertical shift moves the graph up or down. In y = -2 + (1/2) sin 3x, amplitude is 1/2, frequency is 3 (affecting period), and vertical shift is -2.

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Phase Shifts

Period of a Trigonometric Function

The period is the length of one complete cycle of the function. For y = sin(bx), the period is 2π divided by |b|. Here, with b = 3, the period is 2π/3, so a two-period interval spans 4π/3, which guides the domain over which to graph the function.

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Percorso guidato

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Period of Sine and Cosine Functions

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