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Chapter 4: Problem 60

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

Short Answer

Expert verified
The amplitude of the graph is 3, the period is \(\frac{\pi}{3}\), and the phase shift is -\(\pi\)/6. The graph starts from the maximum point at the phase shift, reaches the minimum at the mid-period point, and then goes back to the maximum at the end of the period.

Step by step solution

01

Determine the Amplitude

The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine function. Here, the coefficient is -3, and the absolute value of -3 is 3. So, the amplitude is 3.
02

Determine the Period

The period of a cosine function is calculated as \(\frac{2\pi}{|b|}\), where 'b' is the coefficient of x inside the cosine function. In this function, 'b' is 6. Therefore, the period is \(\frac{2\pi}{6} = \frac{\pi}{3}\).
03

Determine the Phase Shift

The phase shift of a cosine function is the value of 'c' in the equation \(y = a \cos (bx + c)\). Here, 'c' is \(\pi\). The phase shift is -c/b = -\(\pi\)/6 as the graph of the function moves to the right for positive 'c' and to the left for negative 'c'.
04

Sketch the Graph

Sketching the graph involves understanding the general shape of the cosine function (which starts at the peak and then descends into a trough before rising up to the peak again) and then modifying this shape according to the amplitude, period, and phase shift. Start with the maximum point at the phase shift on the x-axis, and show the function repeating every period along the x-axis. Because this is a negative cosine function, it will start descending from the maximum point and the amplitude will determine the maximum and minimum points of the function (3 and -3 in this case).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
The amplitude of a trigonometric function such as cosine dictates the height or depth of the oscillations from the midline. It's the absolute value of the coefficient in front of the function. In the given function \( y = -3 \cos(6x + \pi) \), the coefficient is -3.
Simply take the absolute value, and the amplitude is therefore 3.
This means the cosine wave will reach a maximum of 3 and a minimum of -3, centered around the horizontal axis of the graph.
Understanding amplitude is crucial because it tells us how "tall" or "short" the waves will appear in the graph.
Period
The period of a trigonometric function indicates the distance along the x-axis that the function takes to complete one full cycle. For the cosine function, the formula for the period is \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \) inside the argument.
In the exercise, for the function \( y = -3 \cos(6x + \pi) \), \( b \) is 6.
Therefore, the period is calculated as \( \frac{2\pi}{6} = \frac{\pi}{3} \).
This tells us that every \( \frac{\pi}{3} \) units on the x-axis, the cosine graph repeats its entire shape, starting over with the same pattern.
Phase Shift
Phase shift describes the horizontal movement of a trigonometric graph along the x-axis. It happens because of the constant added or subtracted within the function. In \( y = a \cos(bx + c) \), the phase shift is calculated as \(- \frac{c}{b}\).
For \( y = -3 \cos(6x + \pi) \), \( c \) is \( \pi \), and \( b \) is 6.
Thus, the phase shift is \(- \frac{\pi}{6}\).
This negative value tells us the graph shifts \( \frac{\pi}{6} \) units to the left. Understanding the phase shift helps in placing the waveform correctly on the graph by knowing where the peaks and troughs start.
Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the shape and position dictated by their characteristics such as amplitude, period, and phase shift. For the function \( y = -3 \cos(6x + \pi) \):
  • Start by placing the first maximum at the phase-shifted position along the x-axis.
  • The function begins with a descending wave because it's a negative cosine function.
  • Each full cycle consists of this descent to the minimum value (in this case, -3), followed by an ascent back to the maximum (3).
  • This pattern will repeat every \( \frac{\pi}{3} \) units along the x-axis.
By addressing these key factors, we can accurately sketch the graph and see how it repeats in waves both left and right from the initial position.

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Most popular questions from this chapter

Use a graph to solve the equation on the interval \([-2 \pi, 2 \pi]\). $$ \sec x=2 $$

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