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Chapter 4: Problem 25

Describe the relationship between the graphs of \(f\) and \(g .\) Consider amplitude, period, and shifts. $$\begin{aligned} &f(x)=\sin 2 x\\\ &g(x)=3+\sin 2 x \end{aligned}$$

Short Answer

Expert verified
Both functions \(f(x) = \sin 2x\) and \(g(x) = 3 + \sin 2x\) have the same amplitude of 1 and period of \(\pi\). However, \(g(x)\) has a vertical shift of 3 units upwards, while \(f(x)\) has no vertical shift.

Step by step solution

01

Identify the Amplitude of Both Functions

For a sine function of the form \(A \sin (Bx+C)\), the amplitude is given by the coefficient of the sine factor, which is the absolute value of A. In this case the amplitude of \(f(x)\) and \(g(x)\) is 1 and 1 respectively, because the coefficient of the \(\sin(2x)\) term in both functions is 1.
02

Identify the Period of Both Functions

The period is indirectly given by the coefficient of \(x\) inside the sine function. The period of a sine function in general form \(A \sin (Bx+C)\) is given by \(2\pi/B\). Thus, the period for \(f(x)\) and \(g(x)\) is \(2\pi/2 = \pi\).
03

Identify Vertical Shifts of Both Functions

A vertical shift is present when there is a constant term being added to or subtracted from the sine function. \(f(x)\) has no vertical shift since there is no constant added or subtracted from \(\sin(2x)\). However, \(g(x)\) has a vertical shift of 3 units upwards due to the +3 term.

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Most popular questions from this chapter

For the simple harmonic motion described by the trigonometric function, find (a) the maximum displacement, (b) the frequency, (c) the value of \(d\) when \(t=5,\) and (d) the least positive value of \(t\) for which \(d=0 .\) Use a graphing utility to verify your results. $$d=\frac{1}{4} \sin 6 \pi t$$

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