Question

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

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

- 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.

- 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: \( D = 7 \)

- 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.

Key concepts

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.