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Chapter 4: Problem 58

Sketch the graph of the function. (Include two full periods.) $$y=-3 \cos (6 x+\pi)$$

Short Answer

Expert verified
The graph of the function \(y=-3 \cos (6 x+\pi)\) is a cosine function reflected across the x-axis with an amplitude of 3, a period of \(\frac{\pi}{3}\), and a phase shift of \(\frac{\pi}{6}\) to the left. After sketching the first period, repeat the same pattern to sketch the second period. Ensure to label the amplitude, period and phase shift correctly.

Step by step solution

01

Identify Amplitude, Frequency and Phase Shift

First identify the amplitude, frequency, and phase shift from the given function. The amplitude is the absolute value of the coefficient of the cosine function, which is |-3| = 3. The frequency is the coefficient of \(x\) inside the cosine function, which is 6. The phase shift can be calculated by dividing the constant inside the cosine function by the frequency, which gives us \(\frac{\pi}{6}\) to the left.
02

Find the Period

Next, find the period of the function. For the standard cosine function, the period is \(2\pi\). However, when there is a frequency of \(b\), the period is given by \(\frac{2\pi}{|b|}\). Here, \(b=6\), so the period is \(\frac{2\pi}{6}\) or \(\frac{\pi}{3}\). Note that the period is the interval on which the function completes one full cycle.
03

Sketch the Graph

Now, you can sketch the graph of the function. Begin from the phase shift, which is \(\frac{\pi}{6}\) to the left. Recall that the cosine function starts at the maximum value, moves downwards to the minimum and then back up to the maximum in one period. However, since our function has a negative amplitude, we begin from the minimum value, move upwards to the maximum and then back down. Start at the point \((-\frac{\pi}{6}, -3)\) (due to the phase shift and negative amplitude), and draw the function completing 2 full periods which involves a sine wave going from \(y=-3\) to \(y=3\) twice over the interval \(x = -\frac{\pi}{6}\) to \(x = -\frac{\pi}{6} + 2 \times \frac{\pi}{3}\).
04

Label Key Features

Finally, label the amplitude, period, and phase shift on your sketch. This makes your work easier to understand.

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Most popular questions from this chapter

Use a graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\). (a) As \(x \rightarrow 0^{+},\) the value of \(f(x) \rightarrow\) (b) As \(x \rightarrow 0^{-},\) the value of \(f(x) \rightarrow\) (c) As \(x \rightarrow \pi^{+},\) the value of \(f(x) \rightarrow\) (d) As \(x \rightarrow \pi^{-},\) the value of \(f(x) \rightarrow\) $$f(x)=\cot x$$

The circular blade on a saw rotates at 5000 revolutions per minute. (a) Find the angular speed of the blade in radians per minute. (b) The blade has a diameter of \(7 \frac{1}{4}\) inches. Find the linear speed of a blade tip.

A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is 2 feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute.

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow-1^{+}, \text {the value of } \arccos x \rightarrow\text { _____ } .$$

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. $$d=9 \cos \frac{6 \pi}{5} t$$
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