Determine the simplest form of an equation for each graph. Choose b \u003e 0, and include no phase shifts.

Ch. 4 - Graphs of the Circular Functions

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Lial 12th Edition

Ch. 4 - Graphs of the Circular Functions

문제 43

5장, 문제 43

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

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

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