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Ch. 4 - Graphs of the Circular Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 4 - Graphs of the Circular FunctionsProblem 39
Chapter 5, Problem 39

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts. (Midpoints and quarter points are identified by dots.)


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Identify the type of trigonometric function represented by the graph. Since midpoints and quarter points are marked, check if the graph resembles a sine or cosine wave by noting where it starts and its shape.
Determine the amplitude (a) by measuring the vertical distance from the midline (equilibrium position) to the maximum or minimum point of the graph. This gives the coefficient 'a' in the function \(y = a \sin(bx)\) or \(y = a \cos(bx)\).
Find the period (T) of the function by measuring the horizontal length of one complete cycle of the wave. Use the relationship between period and frequency: \(b = \frac{2\pi}{T}\), where \(b > 0\) is the frequency multiplier inside the function argument.
Since the problem specifies no phase shifts, ensure the function starts at the appropriate point for sine or cosine without horizontal translation. For example, sine typically starts at the midline going upward, cosine starts at the maximum.
Write the equation in the form \(y = a \sin(bx)\) or \(y = a \cos(bx)\) using the values of amplitude 'a' and frequency 'b' found, ensuring \(b > 0\) and no phase shift term is included.

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Key Concepts

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

Basic Form of Sine and Cosine Functions

The general equations for sine and cosine functions are y = a sin(bx + c) + d and y = a cos(bx + c) + d, where a is amplitude, b affects the period, c is the phase shift, and d is the vertical shift. Understanding these parameters helps in matching the graph to its equation.
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Graph of Sine and Cosine Function

Period and Frequency of Trigonometric Functions

The period of sine and cosine functions is given by 2π divided by the coefficient b (period = 2π/b). Identifying the period from the graph allows determination of b, which controls how many cycles occur over a given interval.
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Period of Sine and Cosine Functions

Phase Shift and Midpoint Identification

Phase shift c translates the graph horizontally. Since the question specifies no phase shifts, the function should start at standard positions. Using midpoints and quarter points marked on the graph helps confirm the function's alignment and ensures c = 0.
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Phase Shifts
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