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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 1
Capitolo 5, Problema 1

Fill in the blank(s) to correctly complete each sentence.
The graph of y = sin (x + π/4) is obtained by shifting the graph of y = sin x ______ unit(s) to the ________ (right/left).

Guida verificata passo dopo passo
1
Recall that the function y = sin(x + c) represents a horizontal shift of the basic sine function y = sin x, where c is a constant.
If the function is y = sin(x + \(\frac{\pi}{4}\)), the graph is shifted horizontally by \(\frac{\pi}{4}\) units.
Since the argument inside the sine function is (x + \(\frac{\pi}{4}\)), this corresponds to a shift to the left by \(\frac{\pi}{4}\) units (because adding inside the function shifts the graph left).
Therefore, the graph of y = sin(x + \(\frac{\pi}{4}\)) is obtained by shifting the graph of y = sin x \(\frac{\pi}{4}\) unit(s) to the left.
This is a standard property of function transformations: y = sin(x + c) shifts the graph left by c units, and y = sin(x - c) shifts it right by c units.

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Phase Shift in Trigonometric Functions

Phase shift refers to the horizontal translation of a trigonometric graph caused by adding or subtracting a constant inside the function's argument. For y = sin(x + π/4), the graph shifts horizontally by π/4 units. The sign inside the function determines the direction of the shift.
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6:31
Phase Shifts

Graph of the Sine Function

The sine function y = sin x is periodic with period 2π and oscillates between -1 and 1. Understanding its basic shape and key points helps in visualizing transformations like shifts, stretches, and reflections applied to the graph.
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5:53
Graph of Sine and Cosine Function

Effect of Adding a Positive Constant Inside the Function Argument

Adding a positive constant inside the argument of sine, as in sin(x + c), shifts the graph to the left by c units. This is because the input values that produce the same output occur earlier on the x-axis, effectively moving the graph leftward.
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3:30
Example 3
Pratica correlata
Domanda del libro di testo

An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.

𝒮(t) = 5 cos 2t

What is the frequency?

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Domanda del libro di testo

Fill in the blank(s) to correctly complete each sentence.

The graph of y = cos (x - π/6) is obtained by shifting the graph of y = cos x ______ unit(s) to the ________ (right/left).

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Domanda del libro di testo

An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.

𝒮(t) = 5 cos 2t

What is the amplitude of this motion?

700
views
Domanda del libro di testo

An object in simple harmonic motion has position function s(t), in inches, from an equilibrium point, as follows, where t is time in seconds.

𝒮(t) = 5 cos 2t

What is the period of this motion?

647
views