Textbook Question
Graph each function over a two-period interval.
y = cos (x - π/2 )
Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 5, Problem 35
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 3 cos [π/2 (x - ½)]
Verified step by step guidance1
Identify the general form of the cosine function: \(y = A \cos(B(x - C)) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) is the vertical translation.
Find the amplitude \(A\) by looking at the coefficient in front of the cosine function. In this case, \(A = 3\), so the amplitude is \(3\).
Determine the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B = \frac{\pi}{2}\), so substitute this value to find the period.
Identify the phase shift by looking at the expression inside the cosine function: \(x - \frac{1}{2}\). The phase shift is \(C = \frac{1}{2}\), which means the graph shifts to the right by \(\frac{1}{2}\) units.
Check for vertical translation \(D\). Since there is no constant added or subtracted outside the cosine function, the vertical translation is \(0\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude of a Trigonometric Function
Amplitude is the maximum absolute value of a trigonometric function from its midline. For functions like y = a cos(bx + c), the amplitude is |a|, representing the height of the wave peaks above or below the central axis.
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Introduction to Trigonometric Functions
Period of a Trigonometric Function
The period is the length of one complete cycle of the function. For y = cos(bx), the period is calculated as (2π) / |b|. It determines how frequently the wave repeats along the x-axis.
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5:33Period of Sine and Cosine Functions
Phase Shift and Vertical Translation
Phase shift is the horizontal shift of the function, found by solving (bx + c) = 0 for x, often expressed as -c/b. Vertical translation is the upward or downward shift of the midline, represented by a constant added or subtracted outside the function.
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6:31Phase Shifts
Related Practice
Textbook Question
Graph each function over a two-period interval.
y = 1 - 2 cot [2(x + π/2)]
Textbook Question
Graph each function over a two-period interval.
y= -1 + (1/2) cot (2x - 3π)
Textbook Question
Graph each function over a two-period interval.
y = -1 + 2 tan x
Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 2 - sin(3x - π/5)
Textbook Question
For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = -¼ cos (½ x + π/2)