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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 44
Chapter 5, Problem 44

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.
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Verified step by step guidance
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Identify the type of trigonometric function represented by the graph (sine or cosine) by observing where the graph starts at x = 0. If it starts at a maximum or minimum, it is likely a cosine function; if it starts at zero and goes upward, it is likely a sine function.
Determine the amplitude (a) by measuring the distance from the midline (usually y = 0) to the maximum or minimum value of the graph. The amplitude is the absolute value of this distance.
Find the period (T) of the function by measuring the length of one complete cycle on the x-axis. Use the formula for the period of sine or cosine: \(T = \frac{2\pi}{b}\), where \(b\) is the frequency parameter you need to find.
Solve for \(b\) using the period: rearrange the formula to \(b = \frac{2\pi}{T}\). Since the problem states \(b > 0\), take the positive value.
Write the equation in the form \(y = a \sin(bx)\) or \(y = a \cos(bx)\) without any phase shifts, using the values of \(a\) and \(b\) found in the previous steps.

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Key Concepts

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

Basic Form of Trigonometric Functions

Trigonometric functions like sine and cosine are generally expressed as y = a sin(bx + c) + d or y = a cos(bx + c) + d, where 'a' is amplitude, 'b' affects period, 'c' is phase shift, and 'd' is vertical shift. Understanding this form helps in identifying parameters from a graph.
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Introduction to Trigonometric Functions

Amplitude and Period

Amplitude is the height from the midline to the peak of the wave, given by |a|, and period is the length of one complete cycle, calculated as 2π/b. Recognizing these from the graph allows determination of 'a' and 'b' values in the equation.
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Period of Sine and Cosine Functions

Phase Shift and Horizontal Translation

Phase shift represents horizontal shifts of the graph and is given by -c/b in the function y = a sin(bx + c). Since the question specifies no phase shifts, the equation should have c = 0, meaning the graph starts at the standard position without horizontal translation.
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Phase Shifts
Related Practice
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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Textbook Question

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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Textbook Question

Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.

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Textbook Question

Graph each function over a one-period interval. See Example 3.

y = (3/2) sin [2(x + π/4)]

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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