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Chapter 4: Problem 36

Determine the amplitude and period of each function. Then graph one period of the function. $$y=\cos 4 x$$

Short Answer

Expert verified
The amplitude of the function \(y=\cos 4x\) is 1, and the period is \(\pi/2\). The function is graphed over one period from 0 to \(\pi/2\), starting at the maximum amplitude, crossing the x-axis at \(\pi/8\) and reaching the minimum amplitude at \(\pi/4\).

Step by step solution

01

Determine the Amplitude

The amplitude of a cosine function is given by the absolute value of A, in the general form of the function \(y=A\cos(Bx+C)+D\). From the function \(y=\cos 4x\), the coefficient of the cosine function (A) is 1. Thus, the amplitude is \(|1|=1\).
02

Determine the Period

To find the period of the cosine function, we use the formula \(T=2\pi/B\), where B is the coefficient of the argument of the cosine function. Here in the function \(y=\cos 4x\), B is 4. Therefore, substituting B = 4 into the formula, the period is \(T=2\pi/4=\pi/2\).
03

Graph the Function

Plotting the function over one period: Start by drawing the x-axis from 0 to \(\pi/2\). Then, plot the points (0,1), \(\pi/8\), 0 and \(\pi/4\), -1, denoting the maximum, zero, and minimum points of the cosine function in a period respectively. Finally, connect the points with a smooth curve, noting that the cosine function starts from the peak, goes through zero at \(\pi/8\), and ends at the minimum point at \(\pi/4\).

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Most popular questions from this chapter

Graph \(y=\sin \frac{1}{x}\) in a [-0.2,0.2,0.01] by [-1.2,1.2,0.01] viewing rectangle. What is happening as \(x\) approaches 0 from the left or the right? Explain this behavior.

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\frac{1}{2} \tan (\pi x+1)$$

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-2.5 \sin \frac{\pi}{3} x \text { and } y=-2.5 \csc \frac{\pi}{3} x$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \sin (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

Carbon dioxide particles in our atmosphere trap heat and raise the planet's temperature. Even if all greenhousegas emissions miraculously ended today, the planet would continue to warm through the rest of the century because of the amount of carbon we have already added to the atmosphere. Carbon dioxide accounts for about half of global warming. The function $$y=2.5 \sin 2 \pi x+0.0216 x^{2}+0.654 x+316$$ models carbon dioxide concentration, \(y,\) in parts per million, where \(x=0\) represents January \(1960 ; x=\frac{1}{12},\) February \(1960 ; x=\frac{2}{12},\) March \(1960 ; \ldots, x=1,\) January \(1961 ; x=\frac{13}{12}\) February \(1961 ;\) and so on. Use a graphing utility to graph the function in a [30,48,5] by [310,420,5] viewing rectangle. Describe what the graph reveals about carbon dioxide concentration from 1990 through 2008
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