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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 16
Capitolo 5, Problema 16

Match each function with its graph in choices A–I. (One choice will not be used.)
y = -1 + cos x


A. <IMAGE> B. <IMAGE> C. <IMAGE>


D. <IMAGE> E. <IMAGE> F. <IMAGE>


G. <IMAGE> H. <IMAGE> I. <IMAGE>

Guida verificata passo dopo passo
1
Identify the base function in the given equation: here, it is the cosine function, \(\cos x\).
Note the transformation applied to the base function: the function is shifted vertically by \(-1\), so the graph of \(y = \cos x\) is moved down by 1 unit.
Recall the key characteristics of \(y = \cos x\): it has a maximum value of 1, a minimum value of -1, and a midline at \(y=0\). After the vertical shift, the new midline will be at \(y = -1\).
Analyze the amplitude and period: the amplitude remains 1 (since there is no coefficient multiplying \(\cos x\)), and the period remains \(2\pi\).
Match the graph that shows a cosine wave with its midline at \(y = -1\), maximum at 0, minimum at \(-2\), and the usual cosine wave shape starting at a maximum point when \(x=0\).

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Graphing Transformations of Trigonometric Functions

Understanding how vertical shifts, reflections, and stretches affect the graph of trigonometric functions is essential. For example, y = -1 + cos x represents a cosine graph shifted down by 1 unit. Recognizing these transformations helps match the function to its correct graph.
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Percorso guidato
4:22
Domain and Range of Function Transformations

Properties of the Cosine Function

The cosine function has a period of 2π, an amplitude of 1, and ranges between -1 and 1. Knowing its shape, maxima, minima, and zeros allows you to identify its graph and distinguish it from sine or other trigonometric functions.
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Graph of Sine and Cosine Function

Function and Graph Matching Techniques

Matching functions to graphs involves analyzing key features such as amplitude, period, phase shift, and vertical shift. Comparing these features systematically enables accurate identification of the correct graph among multiple choices.
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Percorso guidato
5:53
Graph of Sine and Cosine Function
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Graph each function over a one-period interval.

y = 2 tan (¼ x)

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Match each function in Column I with the appropriate description in Column II.


I

y = 3 sin(2x - 4)


II

A. amplitude = 2, period = π/2, phase shift = ¾

B. amplitude = 3, period = π, phase shift = 2

C. amplitude = 4, period = 2π/3, phase shift = ⅔

D. amplitude = 2, period = 2π/3, phase shift = 4⁄3

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For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = 1/3 tan (3x - π/3)

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For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.

y = cot (x/2 + 3π/4)

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Graph each function over a one-period interval.

y = cot (3x)

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Graph each function over a one-period interval. See Examples 1–3.

y = 2 tan x

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