Domanda del libro di testo
Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.
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Ch. 4 - Graphs of the Circular Functions
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891
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TOC.tableOfContents
- 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
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Capitolo 5, Problema 43
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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Guida verificata passo dopo passo1
Identify the type of trigonometric function represented by the graph (sine or cosine) by observing the starting point of the graph at x = 0. For example, if the graph starts at the midline going upward, it is likely a sine function; if it starts at a maximum or minimum, it is likely a cosine function.
Determine the amplitude (a) by measuring the vertical distance from the midline (horizontal center line) to the maximum or minimum point of the graph.
Find the period (T) by measuring the horizontal length of one complete cycle of the graph. Use the formula for the period of sine or cosine: \(T = \frac{2\pi}{b}\), and solve for \(b\) as \(b = \frac{2\pi}{T}\).
Since the problem specifies no phase shifts, set the phase shift \(c = 0\). Write the general form of the function as \(y = a \sin(bx)\) or \(y = a \cos(bx)\) depending on the function type identified in step 1.
Verify the function by checking key points such as midpoints and quarter points (where the function reaches zero, maximum, or minimum) to ensure the equation matches the graph's behavior without any horizontal shifts.
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Trigonometric Function Forms
Understanding the standard forms of sine and cosine functions, such as y = a sin(bx) or y = a cos(bx), is essential. These forms include amplitude (a), frequency (b), and phase shift, which define the shape and position of the graph. Recognizing these helps in writing the equation from the graph.
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Percorso guidato
6:04
Introduction to Trigonometric Functions
Amplitude and Period
Amplitude is the height from the midline to the peak of the wave, indicating the value of 'a'. The period is the length of one complete cycle, calculated as 2π divided by the frequency 'b'. Identifying these from the graph allows determination of 'a' and 'b' in the equation.
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Percorso guidato
5:33Period of Sine and Cosine Functions
Phase Shift and Midline
Phase shift refers to horizontal shifts of the graph, but the question specifies no phase shifts, simplifying the equation. The midline is the horizontal axis around which the function oscillates, usually y=0 unless shifted vertically. Recognizing these helps in writing the simplest form without additional shifts.
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Percorso guidato
6:31Phase Shifts
Pratica correlata
Domanda del libro di testo
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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841
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Domanda del libro di testo
Determine the simplest form of an equation for each graph. Choose b > 0, and include no phase shifts.
<IMAGE>
634
views
Domanda del libro di testo
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.)
<IMAGE>
796
views
Domanda del libro di testo
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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769
views
Domanda del libro di testo
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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667
views