Determine an equation of the form y = a cos bx or y = a sin bx, where b \u003e 0, for the given graph. See Example 6.

Ch. 4 - Graphs of the Circular Functions

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Ch. 4 - Graphs of the Circular Functions

문제 4.42

5장, 문제 4.42

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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검증된 단계별 안내

Identify the type of trigonometric function: Determine whether the graph resembles a sine or cosine function based on its shape and starting point.

Determine the amplitude 'a': Measure the vertical distance from the midline of the graph to a peak or trough. This value is the amplitude.

Determine the period of the function: Measure the horizontal distance required for the graph to complete one full cycle.

Calculate the value of 'b': Use the formula for the period of a sine or cosine function, which is \( \frac{2\pi}{b} \), and solve for 'b'.

Write the equation: Substitute the values of 'a' and 'b' into the form \( y = a \cos(bx) \) or \( y = a \sin(bx) \), depending on the function type identified in step 1.

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주요 개념

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

Amplitude

Amplitude refers to the maximum height of a wave from its central axis. In the equations y = a cos(bx) or y = a sin(bx), the value 'a' represents the amplitude. It determines how far the graph stretches vertically from the midline, affecting the overall height of the peaks and depth of the troughs.

Period

The period of a trigonometric function is the distance along the x-axis required for the function to complete one full cycle. In the equations y = a cos(bx) or y = a sin(bx), the period is calculated as 2π/b. This concept is crucial for understanding how frequently the wave oscillates and is essential for matching the graph to the correct function.

Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric graph. It occurs when the function is adjusted by adding or subtracting a constant inside the argument of the sine or cosine function. Understanding phase shift is important for accurately positioning the graph along the x-axis, which can be necessary to match the given graph in the problem.

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>