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Chapter 5: Problem 12

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

Short Answer

Expert verified
Amplitude: 4, Period: 2\( \pi \), Phase Shift: 0

Step by step solution

01

Identify the amplitude

The amplitude is the coefficient of \(\text{cos} x\). In the function \(-4 \text{cos} x\), the coefficient is \(-4\). The amplitude is always taken as the absolute value.\[Amplitude = | -4 | = 4\]
02

Determine the period

The period of a cosine function \(\cos bx\) is given by \[\frac{2 \pi}{b}\]. In the function given, \(\text{cos} x\), \(b = 1\). Thus,\[Period = \frac{2 \pi}{1} = 2 \pi\]
03

Find the phase shift

For the general cosine function \(y = \cos(bx - c)\), the phase shift is given by \[ \frac{c}{b} \]. In the given function, \(y = -4 \cos x\), there is no \(c\) term (equivalent to \(c=0\)), so the phase shift is\[Phase Shift = \frac{0}{1} = 0\]

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Key Concepts

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

Understanding Amplitude
Amplitude represents the height of the wave from its central axis to the peak. In simpler terms, it's how tall the wave is. For the trigonometric function provided, \(y = -4 \cos x\), the amplitude can be identified by looking at the coefficient in front of the cosine function.

In this case, the coefficient is \(-4\). When determining amplitude, we always take the absolute value of this coefficient.

Thus, the amplitude is \[Amplitude = | -4 | = 4 \]

It's important to note that the negative sign in front of the 4 influences the direction of the wave (whether it starts up or down), but not the amplitude's size itself.
Exploring the Period
The period of a trigonometric function tells us how long it takes for the wave to complete one full cycle. For cosine and sine functions, this is influenced by the value next to the variable inside the cosine or sine function.

For the function \(y = -4 \cos x\), we can identify the period by looking at the formula for the cosine function. The general formula is \(\text{cos} (bx)\). The period is calculated using the following formula:

\[Period = \frac{2 \pi}{b}\]

Where \(b\) is the coefficient of \(x\). In our function, \(b = 1\), so the period becomes:

\[Period = \frac{2 \pi}{1} = 2 \pi\]

This implies our cosine wave will repeat every \[2 \pi\] units.
Phase Shift Explained
The phase shift in a trigonometric function dictates how much the function is horizontally shifted from its usual position. For a cosine function, the phase shift can be determined from the general function of the form \(y = \cos (bx - c)\).

The horizontal shift is given by the formula:

\[Phase \Shift = \frac{c}{b}\]

For the provided function \(y = -4 \cos x\), there is no \(c\) term visible, which is equivalent to having \(c = 0\). Consequently:

\[Phase \Shift = \frac{0}{1} = 0\]

This means there is no horizontal shift in the position of our cosine function. It remains centered at its usual starting angle.

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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. $$y=\sec (\pi x / 2)$$

Write the equation of each curve in its final position. The graph of \(y=\cot (x)\) is shifted \(\pi / 2\) units to the left, reflected in the \(x\) -axis, then translated 1 unit upward.

Solve each problem. Find \(\sin (\alpha),\) given that \(\cos (\alpha)=-4 / 5\) and \(\alpha\) is in quadrant III.

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

$$\text { Solve } \log _{2}(x)-\log _{2}(x+3)=-3$$
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