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Chapter 7: Problem 17

Determine the amplitude, period, and phase shift for the given function. Graph the function over one period. Indicate the \(x\) -intercepts and the coordinates of the highest and lowest points on the graph. $$g(x)=\cos \left(x+\frac{\pi}{3}\right)$$

Short Answer

Expert verified
Amplitude: 1, Period: \(2\pi\), Phase Shift: \(-\frac{\pi}{3}\).

Step by step solution

01

Understand the Trigonometric Function

We are given the cosine function in the form \[g(x) = a \cos(bx + c) + d\]For this problem, \(a = 1\), \(b = 1\), \(c = \frac{\pi}{3}\), and \(d = 0\). The function can thus be rewritten as \[g(x) = \cos\left(x + \frac{\pi}{3}\right)\]No vertical shift since \(d = 0\).
02

Determine the Amplitude

The amplitude of a cosine function \(a \cos(bx + c) + d\) is given by the absolute value of \(a\). Here, \(a = 1\), so the amplitude is 1.
03

Calculate the Period

The period of the cosine function is given by \[\frac{2\pi}{|b|}\]In this case, \(b = 1\), so the period is \[\frac{2\pi}{1} = 2\pi\]Thus, the function completes one cycle over an interval of \([0, 2\pi]\).
04

Find the Phase Shift

The phase shift of a cosine function \(a \cos(bx + c) + d\) is \[-\frac{c}{b}\]In this case, \(c = \frac{\pi}{3}\) and \(b = 1\), so the phase shift is \[-\frac{\pi}{3}\]This means the graph is shifted \(\frac{\pi}{3}\) units to the left.
05

Graph the Function

The cosine function starts at its maximum if not phase-shifted. Shift this by \(-\frac{\pi}{3}\) units to the left. Sketch the graph starting at \((x + \frac{\pi}{3} = 0)\), and mark one cycle endpoint at \(x + \frac{\pi}{3} = 2\pi\).
06

Identify Key Points

1. Maximum Point: Since the amplitude is 1, the maximum value is 1 at \(x = -\frac{\pi}{3}\).2. Minimum Point: The minimum value is -1, midway through the period at \(x = \frac{3\pi}{3}\). 3. X-Intercepts: Find points where \(\cos\left(x + \frac{\pi}{3}\right) = 0\), which occur at \[x = \frac{\pi}{6}, \frac{5\pi}{6}.\]All are over one full cycle between \(-\frac{\pi}{3}\) to \(\frac{5\pi}{3}\).

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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 is a measure of how much the function waves above and below its central axis. Specifically, in the function form \(a \cos(bx + c) + d\), the amplitude is determined by \(a\). This essentially represents the peak (maximum) value it achieves from the center of the wave.
In simpler terms, for our function \(g(x) = \cos \left(x+\frac{\pi}{3}\right)\):
  • The amplitude is determined by the coefficient of the cosine function, here it is \(1\).
  • This means the wave oscillates 1 unit above and below its central position, which is the x-axis in this case.
  • Thus, the highest point is \(1\) and the lowest point is \(-1\).
The amplitude is a critical factor because it helps define the vertical stretch of the graph, contributing to the strength or loudness of its "sound" if we were to imagine it as an audio wave.
Period
The period of a trigonometric function like cosine or sine indicates the length of one complete cycle of the wave before it repeats itself. It's computed using the formula \(\frac{2\pi}{|b|}\) for functions of the form \(a \cos(bx + c) + d\).
In our function \(g(x) = \cos \left(x+\frac{\pi}{3}\right)\):
  • \(b\) is \(1\), making the calculation for the period straightforward: \(\frac{2\pi}{1} = 2\pi\).
  • This means the wave completes one cycle every \(2\pi\) units along the x-axis.
  • The function's behavior repeats itself once every \(2\pi\) units; thus, if you started plotting from one peak, you'd reach the next peak exactly \(2\pi\) units later.
The period is essential because it determines how "packed" or "stretched out" the waves appear on the graph. A smaller period packs more waves into less space, while a larger period stretches the waves.
Phase Shift
Phase shift describes the horizontal translation of the trigonometric graph along the x-axis. For functions in the form \(a \cos(bx + c) + d\), the phase shift is calculated as \(-\frac{c}{b}\).
Consider our function \(g(x) = \cos \left(x+\frac{\pi}{3}\right)\):
  • The values involved are \(c = \frac{\pi}{3}\) and \(b = 1\).
  • The phase shift calculation is \(-\frac{\frac{\pi}{3}}{1} = -\frac{\pi}{3}\).
  • This means the graph of the function has shifted \(\frac{\pi}{3}\) units to the left.
Understanding the phase shift helps in placing the wave correctly on the x-axis. It aligns the starting point of the trigonometric cycle to match the values we see in the function. For instance, instead of starting at \((0,1)\), the cosine wave now begins at \(-\frac{\pi}{3}\) due to this leftward shift.

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Most popular questions from this chapter

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