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Chapter 8: Problem 5

find the period and amplitude. $$ y=\frac{1}{2} \cos \pi x $$

Short Answer

Expert verified
The amplitude of the function \(y=\frac{1}{2} \cos \pi x\) is \(\frac{1}{2}\) and the period is \(2\).

Step by step solution

01

Find the Amplitude

The amplitude is indicated by the coefficient of the cosine function. Here, it is \(\frac{1}{2}\). Therefore, the amplitude of this function is \(\frac{1}{2}\).
02

Find the Period

The period can be found using the formula \(T = \frac{2\pi}{B}\), where B is the coefficient of \(x\) in the cosine function. Here, it is \(\pi\). So, Plug \(\pi\) into the given formula to get the period \(T = \frac{2\pi}{\pi}\). Simplifying this fraction gives the period of the function as \(T = 2\).

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Key Concepts

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

Amplitude
In trigonometry, the amplitude of a function is the measure of its height from the center line to the peak (or the trough). Amplitude gives you a sense of how much the function values deviate from the middle. It is always a positive value.
For the cosine function, represented generally as \(y = A \cos(Bx + C) + D\), amplitude is given by the absolute value of \(A\).
In this example, the function is \( y = \frac{1}{2} \cos(\pi x) \). Here, the amplitude \(A\) is \(\frac{1}{2}\), meaning:
  • The furthest distance from the center line that the function reaches is \(\frac{1}{2}\).
  • This indicates both the highest and lowest points from the equilibrium level of the wave.
Think of amplitude as the maximum "high" or "low" a wave reaches. This value gives us insight into how energetic or dampened the wave is.
Period
The period of a trigonometric function is the distance along the x-axis that the function takes to repeat itself. The smaller the period, the quicker it completes a cycle. For the standard cosine function \(y = A \cos(Bx + C) + D\), the period \(T\) is determined using the formula \(T = \frac{2\pi}{B}\).
In the function \( y = \frac{1}{2} \cos(\pi x) \), \(B = \pi\). Therefore, the period \(T\) is calculated as:
  • \(T = \frac{2\pi}{\pi} = 2\)
  • This means the cosine function completes one full cycle every 2 units along the x-axis.
Understanding the period helps visualize how often the wave pattern repeats. It means for every movement of 2 units along the x-axis, the wave pattern of this function will begin its new cycle.
Cosine Function
The cosine function is a key component of trigonometry, representing periodic oscillations. It's one of the basic functions you encounter in trigonometry and is often expressed as \(y = A \cos(Bx + C) + D\). This function gives us the shape of a wave when graphed.
Characteristics of the cosine function include:
  • **Amplitude (\(A\))**: The height from the center line to a peak (or trough), as previously defined.
  • **Period (\(T\))**: Length along the x-axis for a full wave cycle, calculated using \(T = \frac{2\pi}{B}\).
  • **Phase Shift (\(C\))**: Horizontally moves the wave left or right.
  • **Vertical Shift (\(D\))**: Moves the entire graph up or down along the y-axis.
In our given example, the function \(y = \frac{1}{2} \cos(\pi x)\) is a simplified version, having its phase shift \(C\) and vertical shift \(D\) both equal to 0, thus making it centered at the origin without any horizontal or vertical displacement.

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