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Chapter 4: Problem 92

What does a phase shift indicate about the graph of a sine function? How do you determine the phase shift from the function's equation?

Short Answer

Expert verified
A phase shift in a sine function indicates a horizontal displacement or 'start' of the wave. From the function's equation \(y = \sin(B(x-C)) + D\), the phase shift is represented by C. A positive C indicates a shift to the right, and a negative C indicates a shift to the left.

Step by step solution

01

Understanding Sine Function

The basic sine function is given by the equation \(y = \sin(x)\). This function creates a wave pattern that oscillates between -1 and 1. The highest point of the wave in a cycle is called the peak, the lowest point is called the trough and the distance between a point to the equivalent point in the next cycle defines the period of the sine function.
02

Understanding Phase Shift

A phase shift in a sine function is defined as the horizontal displacement of the graph. In simpler terms, it's how much the wave is shifted to the right or left. It basically means where the wave 'starts' from a horizontal perspective.
03

Determining Phase Shift from Equation

The general form of the sine function is \(y = \sin(B(x-C)) + D\), where B affects the period, C is the phase shift, and D is the vertical shift. The phase shift of the function can be determined directly from this equation - it is represented by the coefficient C. If C is positive, the shift is to the right, and if C is negative, the shift is to the left.

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Most popular questions from this chapter

For \(x>0,\) what effect does \(2^{-x}\) in \(y=2^{-x} \sin x\) have on the graph of \(y=\sin x ?\) What kind of behavior can be modeled by a function such as \(y=2^{-x} \sin x ?\)

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