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Chapter 4: Problem 41

Determine the amplitude and period of each function. Then graph one period of the function. $$y=-\frac{1}{2} \cos \frac{\pi}{3} x$$

Short Answer

Expert verified
The amplitude of the function \(y=-\frac{1}{2} \cos \frac{\pi}{3} x\) is \(\frac{1}{2}\) and its period is 6 units. The function is a reflected cosine function with a completed period at x=6.

Step by step solution

01

Find the Amplitude

The amplitude of the function is the absolute value of the coefficient in front of the cosine function. Therefore, the amplitude of the function \(y=-\frac{1}{2} \cos \frac{\pi}{3} x\) is \( |-\frac{1}{2}| = \frac{1}{2} \).
02

Find the period

The period of the function is obtained by dividing \(2 \pi\) by the absolute value of the coefficient in front of the variable 'x'. Therefore, the period of the function \(y=-\frac{1}{2} \cos \frac{\pi}{3} x\) is \( \frac{2 \pi}{\frac{\pi}{3}} = 2\pi \cdot \frac{3}{\pi} = 6 \).
03

Graph the function

This function behaves like the standard cosine function with some transformations. The amplitude modifies the height and the period modifies the length of the function's cycle. Because the coefficient of the cosine function is negative, the function is reflected over the x-axis. The completed period ends at x=6 instead of \(2 \pi\). This means the function starts at (0, -0.5), goes up to (1.5, 0.5), comes back down to (3, -0.5), goes up to (4.5, 0.5) and finally returns back to (6, -0.5) to complete the cycle.

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Most popular questions from this chapter

Use the keys on your calculator or graphing utility for converting an angle in degrees, minutes, and seconds \(\left(D^{\circ} M^{\prime} S^{\prime \prime}\right)\) into decimal form, and vice versa. Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. $$30.42^{\circ}$$

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=3 \cos (2 \pi x+4 \pi)$$

will help you prepare for the material covered in the next section. a. Graph \(y=-3 \cos \frac{x}{2}\) for \(-\pi \leq x \leq 5 \pi\) b. Consider the reciprocal function of \(y=-3 \cos \frac{x}{2}\) namely, \(y=-3 \sec \frac{x}{2} .\) What does your graph from part (a) indicate about this reciprocal function for \(x=-\pi, \pi, 3 \pi,\) and \(5 \pi ?\)

Graph one period of each function. $$y=\left|3 \cos \frac{2 x}{3}\right|$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used a tangent function to model the average monthly temperature of New York City, where \(x=1\) represents January, \(x=2\) represents February, and so on.
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