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Chapter 1: Problem 33

\(f(x)=4 \sin \pi x\) \(g(x)=4 \sin \pi x-3\)

Short Answer

Expert verified
Function \(g(x)\) is the graph of function \(f(x)\), shifted downwards by three units. The period and amplitude remain unchanged.

Step by step solution

01

Understanding function f(x)

The function \(f(x)=4\sin{(\pi x)}\) is a sine function. The coefficient of \(x\) in the argument of the sine function determines the period, while the coefficient outside the sine function determines the amplitude. In this case, the period is \(2/\pi\) (since in general the period of the sine function is found by \(2\pi/\text{coefficient of } x\)), and the amplitude is 4.
02

Understanding function g(x)

The function \(g(x)=4\sin{(\pi x)}-3\) is similar to \(f(x)\), with one modification: it is shifted downwards by 3 units. This is due to the '-3' outside the sine function. This does not change the period or amplitude.
03

Relationship between f(x) and g(x)

\(g(x)\) can be considered as the transformed function of \(f(x)\). It is the graph of \(f(x)\) shifted downwards by 3 units. In terms of their graphs, if the graph of \(f(x)\) lies on any point (a, b), then the graph of \(g(x)\) will lie on the point (a, b-3).

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