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Chapter 4: Problem 19

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitude, period, and shifts. $$\begin{array}{l} f(x)=\sin x \\ g(x)=\sin (x-\pi) \end{array}$$

Short Answer

Expert verified
Both functions \(f(x)=\sin x\) and \(g(x)=\sin (x-\pi)\) have the same amplitude of 1 and period of \(2\pi\), however, \(g(x)=\sin (x-\pi)\) is shifted \(\pi\) units to the right compared to \(f(x)=\sin x\), resulting in a phase shift.

Step by step solution

01

Analyze the function \(f(x)=\sin x\)

First, let's understand the behavior of \(f(x)=\sin x\). The sine function has an amplitude of \(1\), meaning it oscillates between \(1\) and \(-1\). Its period is \(2\pi\), meaning the function repeats every \(2\pi\) units. There are no shifts in this function.
02

Analyze the function \(g(x)=\sin (x-\pi)\)

Next, we look at the function \(g(x)=\sin (x-\pi)\). The amplitude is still \(1\) and the period is still \(2\pi\), as neither of these properties have been affected. The function, however, has been shifted to the right by \(\pi\) units compared to \(f(x)=\sin x\). This is a 'phase shift'.
03

Summarize the relationship

Given that both functions have the same amplitude and period, but \(g(x)=\sin (x-\pi)\) is shifted \(\pi\) units to the right compared to \(f(x)=\sin x\), we can conclude there is a phase shift between the two functions.

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