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Chapter 4: Problem 23

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

Short Answer

Expert verified
The amplitude of both functions is 1. However, \(f(x)\) has a period of \(2\pi\), while \(g(x)\) has a period of \(\pi\). Neither function experiences a phase shift.

Step by step solution

01

Analyze Function f(x)

First, let's accurately analyze the function \(f(x)=\cos x\). The amplitude of \(f(x)\) (which is the height of the graph from its center to a peak or valley) is 1, because the absolute maximum value of the cosine function is 1. The period of \(f(x)\) (one complete graph cycle) is \(2\pi\), because the cosine function completes a full cycle over that range. This function has no phase shift, because there is no horizontal translation in the equation.
02

Analyze Function g(x)

Now let's analyze the function \(g(x)=\cos 2x\). The amplitude of \(g(x)\) is also 1, because there are no vertical transformations in the equation. The period of \(g(x)\) is \(\pi\). This is because multiplying the variable x by 2 in the argument of a trigonometric function halves the period (from \(2\pi\) to \(\pi\), in this case). This function also has no phase shift, because there is no horizontal translation in the equation.
03

Compare f(x) and g(x)

Ultimately, \(f(x)\) and \(g(x)\) have the same amplitude of 1 but different periods. The period of \(g(x)\) is a half that of \(f(x)\). There are no phase shifts in either function

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Most popular questions from this chapter

Graph the functions \(f\) and \(g .\) Use the graphs to make a conjecture about the relationship between the functions. $$f(x)=\cos ^{2} \frac{\pi x}{2}, \quad g(x)=\frac{1}{2}(1+\cos \pi x)$$

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