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Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric FunctionsProblem 27
Chapter 2, Problem 27

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.
y = 3 sin(πx + 2)

Verified step by step guidance
1
Identify the general form of the sine function: \(y = A \sin(Bx + C)\), where \(A\) is the amplitude, \(B\) affects the period, and \(C\) affects the phase shift.
Find the amplitude by taking the absolute value of the coefficient in front of the sine function: \(A = |3|\).
Calculate the period using the formula \(\text{Period} = \frac{2\pi}{|B|}\). Here, \(B\) is the coefficient of \(x\) inside the sine function, which is \(\pi\).
Determine the phase shift using the formula \(\text{Phase shift} = -\frac{C}{B}\). In this case, \(C = 2\) and \(B = \pi\).
To graph one period of the function, start at the phase shift on the x-axis, then plot points over one full period length, using the amplitude to mark the maximum and minimum values of the sine wave.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Amplitude of a Sine Function

Amplitude is the maximum value the sine function attains from its midline, representing the height of its peaks. For y = a sin(bx + c), the amplitude is the absolute value of 'a'. In this case, the amplitude is |3| = 3, indicating the wave oscillates 3 units above and below the midline.
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Period of a Sine Function

The period is the length of one complete cycle of the sine wave. It is calculated as (2π) divided by the absolute value of the coefficient 'b' in y = a sin(bx + c). Here, with b = π, the period is 2π/π = 2, meaning the function repeats every 2 units along the x-axis.
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Phase Shift of a Sine Function

Phase shift refers to the horizontal translation of the sine curve and is found by solving (bx + c) = 0 for x, giving -c/b. For y = 3 sin(πx + 2), the phase shift is -2/π, indicating the graph shifts left by 2/π units. This affects where the wave starts on the x-axis.
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Related Practice
Textbook Question

In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = −4 sin 3π/2 t

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Textbook Question

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ 0.3

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Textbook Question

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.


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Textbook Question

In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

Textbook Question

In Exercises 27–38, use a calculator to find the value of each expression rounded to two decimal places. sin⁻¹ (-0.32)

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Textbook Question

Use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.


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