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Chapter 4: Problem 21

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

Short Answer

Expert verified
Both functions \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\) have the same amplitude of 1 and period of \(\pi\). There are no shifts on either function. However, \(g(x)\) is an upside down version of \(f(x)\) due to the negative coefficient of the cosine function.

Step by step solution

01

Find the amplitude

For both functions, the amplitude is given by the absolute value of the coefficient of the cosine function. In both \(f(x) = \cos 2x\) and \(g(x) = -\cos 2x\), the coefficient is 1 in the first case and -1 in the second case. Hence, the amplitudes are both 1.
02

Determine the period

The period of a cosine function is given by \(\frac{2\pi}{\text{absolute value of the coefficient of } x}\). Here, the coefficients of \(x\) in both the functions are 2, hence the period of both the functions is \(\pi\).
03

Analyze shifts

In this case, there are no shifts along the \(x\) or \(y\) axes for either of the two functions, as there are no added or subtracted constants in either function.
04

Compare the two functions

Looking at the comparisons made - \(f(x)\) and \(g(x)\) are identical in terms of amplitude and period, and neither contains shifts. However, \(f(x)\) and \(g(x)\) differ in terms their reflection about the x-axis since \(g(x)\) is -1 times \(f(x)\), that is, \(g(x)\) is an upside down version of \(f(x)\).

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