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Chapter 10: Problem 13

Determine the amplitude, period, and displacement for each function. Then sketch the graphs of the functions. Check each using a calculator. $$y=30 \cos \left(\frac{1}{3} x+\frac{\pi}{3}\right)$$

Short Answer

Expert verified
Amplitude = 30, Period = 6蟺, Phase Shift = -蟺 (left).

Step by step solution

01

Identify the Amplitude

The amplitude of a cosine function y = A \cos(Bx + C) is given by the absolute value of A. In this case, A = 30, so the amplitude is \(|A| = |30| = 30.\)
02

Determine the Period

The period of a cosine function is calculated using the formula \\[ \text{Period} = \frac{2\pi}{|B|} \] \where B is the coefficient of x. In this function, B = \(\frac{1}{3}\). \So the period is \\[ \text{Period} = \frac{2\pi}{\frac{1}{3}} = 6\pi \] \Thus, the period is \(6\pi\).
03

Calculate the Phase Shift

The phase shift of a cosine function is given by \\[ \text{Phase Shift} = -\frac{C}{B} \] \where C is the horizontal translation. Here, C = \(\frac{\pi}{3}\) and B = \(\frac{1}{3}\), so the phase shift is \\[ \text{Phase Shift} = -\frac{\frac{\pi}{3}}{\frac{1}{3}} = -\pi \] \This indicates that the graph is shifted \(\pi\) units to the left.
04

Sketch the Graph

To sketch the graph, note that the cosine function has an amplitude of 30, a period of \(6\pi\), and a phase shift of \(-\pi\). The baseline (midline) is y = 0. Start sketching from \(-\pi\), and mark the maximum at 30, and the minimum at -30. The complete cycle from the point \(-\pi\) to \(5\pi\) covers the period \(6\pi\). Divide \(6\pi\) into four equal sections for key points.
05

Check Using a Calculator

Use a graphing calculator to plot y = 30 \cos\left(\frac{1}{3}x + \frac{\pi}{3}\right) . Verify the amplitude, period, and phase shift. Ensure the graph matches the calculated values: amplitude of 30, period of \(6\pi\), and a left shift of \(\pi\).

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Key Concepts

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

Cosine Functions
The cosine function is a fundamental trigonometric function. It is often written as \(y = A \cos(Bx + C) + D\) and represents a wave-like pattern that repeats. It's key to understand the structure of this function:
  • \(A\) determines the amplitude, which impacts the wave's height.
  • \(B\) affects the period, dictating how frequently the wave repeats.
  • \(C\) accounts for the horizontal shift, often called the phase shift.
  • \(D\) sets the vertical displacement or baseline of the wave.
Cosine functions describe many natural phenomena and are visualized as oscillations or rotations. The familiar shape of the cosine wave results from these repeating cycles, always symmetrical about the vertical axis.
Amplitude
The amplitude of a cosine function is the measurement from the function's midline to its peak or trough. It dictates the height of the wave. In mathematical terms:
  • The amplitude is expressed as \(|A|\), derived from the formula \(y = A \cos(Bx + C)\).
  • For example, with \(y = 30 \cos\left(\frac{1}{3}x + \frac{\pi}{3}\right)\), the amplitude is \(|30| = 30\).
The amplitude never changes throughout the function's graph. A larger amplitude results in taller waves, whereas a smaller amplitude results in shorter waves. This impacts the wave's maximum and minimum values, which would be \(30\) and \(-30\) respectively in our function.
Period
The period of a cosine function determines how long it takes for the wave to complete one full cycle before repeating. It's an important feature because it identifies the function's frequency.
  • The period of \(y = A \cos(Bx + C)\) is found using \(\text{Period} = \frac{2\pi}{|B|}\).
  • In our function, \(B = \frac{1}{3}\), leading to a period of \(\frac{2\pi}{\frac{1}{3}} = 6\pi\).
This means the wave takes \(6\pi\) units along the x-axis to cycle completely. Understanding the period helps to sketch and anticipate the wave's behavior over different intervals.
Phase Shift
The phase shift of a cosine function determines how the wave is horizontally displaced from the usual position. It helps in aligning the graph to the correct starting point.
  • The phase shift is calculated as \(-\frac{C}{B}\), where \(C\) is the horizontal translation and \(B\) affects the period.
  • In the given function, \(C = \frac{\pi}{3}\) and \(B = \frac{1}{3}\), resulting in a phase shift of \(-\pi\).
This indicates that the whole graph shifts one unit of \(\pi\) to the left on the x-axis, helping in correctly positioning the wave pattern on the graph. Knowing the phase shift is crucial for accurate plotting and for understanding how multiple waves can coincide or differ with each other.

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Most popular questions from this chapter

View at least two cycles of the graphs of the given functions on a calculator. $$y=18 \csc \left(3 x-\frac{\pi}{3}\right)$$

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