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Chapter 5: Problem 2

Determine the phase shift and the vertical displacement with respect to \(y=\cos x\) for each function. Sketch a graph of each function. a) \(y=\cos \left(x-30^{\circ}\right)+12\) b) \(y=\cos \left(x-\frac{\pi}{3}\right)\) c) \(y=\cos \left(x+\frac{5 \pi}{6}\right)+16\) d) \(y=4 \cos \left(x+15^{\circ}\right)+3\) e) \(y=4 \cos (x-\pi)+4\) f) \(y=3 \cos \left(2 x-\frac{\pi}{6}\right)+7\)

Short Answer

Expert verified
a) Phase shift: \(30^{\circ}\), Vertical displacement: 12 b) Phase shift: \(\frac{\pi}{3}\), Vertical displacement: 0 c) Phase shift: \(-\frac{5 \pi}{6}\), Vertical displacement: 16 d) Phase shift: \(-15^{\circ}\), Vertical displacement: 3 e) Phase shift: \(\pi\), Vertical displacement: 4 f) Phase shift: \(\frac{\pi}{12}\), Vertical displacement: 7

Step by step solution

01

- Identify the general cosine function form

The standard form of a cosine function is given by: \[ y = A \, \cos(B(x - C)) + D \] where: - \( A \) is the amplitude, - \( B \) affects the period, - \( C \) determines the phase shift, - \( D \) is the vertical displacement.
02

- Identify the phase shift and vertical displacement for each function

Compare each given function with the standard form to identify the values of \( C \) and \( D \): a) \( y = \cos (x - 30^{\circ}) + 12 \) - Phase shift: \( x = C \), so \( C = 30^{\circ} \) - Vertical displacement: \( D = 12 \) b) \( y = \cos (x - \frac{\pi}{3}) \) - Phase shift: \( C = \frac{\pi}{3} \) - Vertical displacement: \( D = 0 \) c) \( y = \cos (x + \frac{5 \pi}{6}) + 16 \) - Phase shift: \( C = -\frac{5 \pi}{6} \) - Vertical displacement: \( D = 16 \) d) \( y = 4 \cos (x + 15^{\circ}) + 3 \) - Phase shift: \( C = -15^{\circ} \) - Vertical displacement: \( D = 3 \) e) \( y = 4 \cos (x - \pi) + 4 \) - Phase shift: \( C = \pi \) - Vertical displacement: \( D = 4 \) f) \( y = 3 \cos (2x - \frac{\pi}{6}) + 7 \) - Phase shift: \( C = \frac{\pi}{12} \) (Note: Divide phase shift by the coefficient of x) - Vertical displacement: \( D = 7 \)
03

- Sketch the graphs

For each function, plot the graph using the identified phase shift and vertical displacement: a) Shift the graph of \( y = \cos x \) right by \( 30^{\circ} \) and up by 12 units. b) Shift the graph of \( y = \cos x \) right by \( \frac{\pi}{3} \) units. c) Shift the graph of \( y = \cos x \) left by \( \frac{5 \pi}{6} \) and up by 16 units. d) Shift the graph of \( y = 4\cos x \) left by \( 15^{\circ} \) and up by 3 units. e) Shift the graph of \( y = 4\cos x \) right by \( \pi \) and up by 4 units. f) Shift the graph of \( y = 3\cos(2x) \) right by \( \frac{\pi}{12} \) and up by 7 units.

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Key Concepts

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

phase shift
The phase shift of a cosine function describes a horizontal shift along the x-axis. If you compare the function to the standard form \( y = A \, \cos(B(x - C)) + D \), you'll notice that the phase shift is determined by the value of \( C \). For example, in the function \( y = \cos(x - 30^{\circ}) + 12 \), the phase shift is \( 30^{\circ} \) to the right. It's crucial to recognize if the phase shift is positive or negative, as it dictates the direction of the shift:
  • Positive \( C \): Shift to the right
  • Negative \( C \): Shift to the left

To find the phase shift in functions involving coefficients in front of \( x \), like \( y = 3 \cos(2x - \frac{\pi}{6}) + 7 \), divide \( C \) by the coefficient. Here, the phase shift is \( \frac{\pi}{12} \).
vertical displacement
Vertical displacement refers to how far the entire function is moved up or down along the y-axis. In the standard cosine function form \( y = A \, \cos(B(x - C)) + D \), it is indicated by \( D \). For instance, if you have the function \( y = \cos(x) + 12 \), it means the whole cosine graph is lifted up by 12 units. Here are some key points:
  • Positive \( D \): Shift upwards
  • Negative \( D \): Shift downwards

In functions with no explicit vertical shift, such as \( y = \cos(x - \frac{\pi}{3}) \), the vertical displacement is \( D = 0 \), meaning there's no vertical movement. If a function like \( y = 4 \cos(x - \pi) + 4 \) is given, the graph moves up by 4 units due to the positive 4.
graphing trigonometric functions
Graphing trigonometric functions utilizes transformations like phase shift, vertical displacement, amplitude, and period adjustments. Start with the standard cosine function \( y = \cos(x) \). Then use the transformations to adjust:
  • Phase Shift: Horizontal shift due to \( C \).
  • Vertical Displacement: Vertical shift due to \( D \).
  • Amplitude: Scale the graph by \( A \).
  • Period: Adjust the period by modifying \( B \).

For example, to graph \( y = 4 \cos(x - \pi) + 4 \), follow these steps:1. Begin with the cosine graph.2. Shift the graph right by \( \pi \).3. Move it up by 4 units.4. Scale the amplitude to 4.
amplitude and period in trigonometry
Amplitude and period are crucial for understanding the shape of trigonometric graphs. The amplitude (\[ A \]) affects the height, while the period (\[ T \]) influences the length of one complete cycle.
  • Amplitude: Given by \( A \) and represents the peak deviation from the centerline (up or down). For example, in \( y = 4 \cos(x) \), the amplitude is 4, so peaks are at 4 and -4.
  • Period: Calculated using \( \frac{2\pi}{B} \). For the function \( y = 3 \cos(2x) \), the period is \( \frac{2\pi}{2} = \pi \).

Knowing these values allows accurate graphing. For instance, if you have \( y = 3 \cos(2x - \frac{\pi}{6}) + 7 \), the amplitude is 3, and the period is \( \pi \), indicating how frequently the function cycles within \( \pi \) units.

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Most popular questions from this chapter

Have you ever wondered how a calculator or computer program evaluates the sine, cosine, or tangent of a given angle? The calculator or computer program approximates these values using a power series. The terms of a power series contain ascending positive integral powers of a variable. The more terms in the series, the more accurate the approximation. With a calculator in radian mode, verify the following for small values of \(x,\) for example, \(x=0.5\). a) \(\tan x=x+\frac{x^{3}}{3}+\frac{2 x^{5}}{15}+\frac{17 x^{7}}{315}\) b) \(\sin x=x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5040}\) c) \(\cos x=1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}\)

The Wave is a spectacular sandstone formation on the slopes of the Coyote Buttes of the Paria Canyon in Northern Arizona. The Wave is made from 190 million-year-old sand dunes that have turned to red rock. Assume that a cycle of the Wave may be approximated using a cosine curve. The maximum height above sea level is 5100 ft and the minimum height is 5000 ft. The beginning of the cycle is at the 1.75 mile mark of the canyon and the end of this cycle is at the 2.75 mile mark. Write an equation that approximates the pattern of the Wave.

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Determine the amplitude of each function. Then, use the language of transformations to describe how each graph is related to the graph of \(y=\sin x\) a) \(y=3 \sin x\) b) \(y=-5 \sin x\) c) \(y=0.15 \sin x\) d) \(y=-\frac{2}{3} \sin x\)
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