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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 46
Chapter 5, Problem 46

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.
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Identify the key characteristics of the given graph, such as amplitude, period, phase shift, and vertical shift. These will help determine the values of \(a\), \(b\), and whether the function is sine or cosine.
Determine the amplitude \(a\) by finding the distance from the midline of the graph to its maximum or minimum value. This is the absolute value of \(a\) in the equation \(y = a \sin(bx)\) or \(y = a \cos(bx)\).
Calculate the period \(T\) of the function by measuring the length of one complete cycle on the x-axis. Use the formula \(T = \frac{2\pi}{b}\) to solve for \(b\), ensuring \(b > 0\).
Decide whether the function is sine or cosine by examining the starting point of the graph at \(x=0\). If the graph starts at a maximum or minimum, it is likely a cosine function; if it starts at the midline going upward or downward, it is likely a sine function.
Write the equation in the form \(y = a \sin(bx)\) or \(y = a \cos(bx)\) using the values of \(a\) and \(b\) found, and include any necessary phase shifts or vertical shifts if the graph is not centered at the origin or midline.

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Key Concepts

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

General Form of Sine and Cosine Functions

The equations y = a sin bx and y = a cos bx represent sinusoidal functions where 'a' is the amplitude and 'b' affects the period. Understanding these forms helps in modeling periodic behavior by adjusting amplitude and frequency to fit a given graph.
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Graph of Sine and Cosine Function

Amplitude and Period of Trigonometric Functions

Amplitude is the maximum value of the function, given by |a|, and period is the length of one complete cycle, calculated as 2π/b. Identifying these from the graph is essential to determine the parameters 'a' and 'b' in the equation.
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Period of Sine and Cosine Functions

Phase Shift and Choosing Between Sine and Cosine

Sine and cosine functions differ by a horizontal shift; cosine starts at a maximum while sine starts at zero. Recognizing the graph's starting point and shape helps decide whether to use sine or cosine and if a phase shift is needed.
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Phase Shifts
Related Practice
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Graph each function over a one-period interval.

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Textbook Question

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.

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views
Textbook Question

Determine an equation of the form y = a cos bx or y = a sin bx, where b > 0, for the given graph. See Example 6.

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