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Chapter 5: Problem 42

Determine the amplitude, period, and phase shift for each function. $$y=-5 \cos (x+4)+\pi$$

Short Answer

Expert verified
Amplitude: 5, Period: \(2\pi\), Phase Shift: -4, Vertical Shift: \(\pi\)

Step by step solution

01

Identify the standard form of the cosine function

The standard form of the cosine function is given by: \[ y = a \cos(bx + c) + d \] where: - \(a\) is the amplitude - \(b\) affects the period - \(c\) affects the phase shift - \(d\) is the vertical shift
02

Determine the amplitude

In the given function, \(y = -5 \cos(x + 4) + \pi\), the amplitude is the absolute value of \(a\). Here, \(a = -5\), so the amplitude is: \[ \text{Amplitude} = |a| = |-5| = 5 \]
03

Determine the period

The period of the cosine function can be found using the formula: \[ \text{Period} = \frac{2\pi}{b} \] In the given function, \(b = 1\), so the period is: \[ \text{Period} = \frac{2\pi}{1} = 2\pi \]
04

Determine the phase shift

The phase shift can be found using the formula: \[ \text{Phase Shift} = -\frac{c}{b} \] In the given function, \(c = 4\) and \(b = 1\), so the phase shift is: \[ \text{Phase Shift} = -\frac{4}{1} = -4 \]
05

Determine the vertical shift

The vertical shift is given by \(d\). In the given function, \(d = \pi\), so the vertical shift is: \[ \text{Vertical Shift} = \pi \]

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Key Concepts

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

Vertical Shift
The vertical shift indicates how the graph of the cosine function moves up or down along the y-axis. This shift is controlled by the value \( d \) in the standard form \( y = a \cos(bx + c) + d \).

For the function given, \( y = -5 \cos(x + 4) + \pi \), the vertical shift is simply the value of \( d \), which is \( \pi \). This means that the entire graph is shifted upward by \( \pi \). Understanding this helps in visualizing where the centerline of the cosine function is.
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Most popular questions from this chapter

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=-\csc \left(\frac{\pi}{2} x+\frac{\pi}{2}\right)$$

Solve each problem. Find \(\cos (\alpha),\) given that \(\sin (\alpha)=-12 / 13\) and \(\alpha\) is in quadrant IV.

Determine the period and sketch at least one cycle of the graph of each function. $$y=\tan (\pi x)$$

Find the approximate value of each expression. Round to four decimal places. $$\tan \left(-44.6^{\circ}\right)$$

Determine the period and range of each function. $$y=\tan (2 x-\pi)+3$$
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