Skip to main content
Ch. 4 - Graphs of the Circular Functions
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoLial 12th EditionCh. 4 - Graphs of the Circular FunctionsProblema 43
Capitolo 5, Problema 43

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

Guida verificata passo dopo passo
1
Identify the general form of the trigonometric function for the graph. Since the problem specifies no phase shifts and b > 0, the general forms to consider are \(y = a \sin(bx)\) or \(y = a \cos(bx)\).
Determine the amplitude \(a\) by measuring the maximum vertical distance from the midline (usually the x-axis) to the peak of the graph. The amplitude is the absolute value of \(a\).
Find the period \(T\) of the graph by measuring the length of one complete cycle along the x-axis. The period is related to \(b\) by the formula \(T = \frac{2\pi}{b}\).
Solve for \(b\) using the period: \(b = \frac{2\pi}{T}\). Since \(b > 0\), take the positive value.
Write the equation using the determined amplitude and \(b\) value, choosing sine or cosine based on the starting point of the graph (e.g., if the graph starts at zero and goes upward, use sine; if it starts at a maximum, use cosine). The final equation will be \(y = a \sin(bx)\) or \(y = a \cos(bx)\) without any phase shift.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

General Form of Trigonometric Functions

Trigonometric functions like sine and cosine can be expressed as y = a sin(bx + c) or y = a cos(bx + c), where 'a' is amplitude, 'b' affects the period, and 'c' is the phase shift. Understanding this form helps in identifying parameters from a graph.
Video consigliato:
Percorso guidato
6:04
Introduction to Trigonometric Functions

Amplitude and Period

Amplitude is the height from the midline to the peak of the wave, given by |a|. The period is the length of one complete cycle, calculated as 2π/b for sine and cosine functions. Recognizing these from the graph is essential to write the equation correctly.
Video consigliato:
Percorso guidato
5:33
Period of Sine and Cosine Functions

Phase Shift and Horizontal Translation

Phase shift refers to the horizontal shift of the graph, determined by the value of 'c' in the function. Since the question specifies no phase shifts, the equation should have c = 0, meaning the graph starts at the standard position without horizontal translation.
Video consigliato:
Percorso guidato
6:31
Phase Shifts
Pratica correlata
Domanda del libro di testo

Graph each function over a two-period interval.

y = sin (x + π/4)

901
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.

<IMAGE>

841
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>

632
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

Graph each function over a one-period interval.

y = -½ cos (πx - π)

815
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.

<IMAGE>

667
views