Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. Give the period and amplinde. $$y=\cos 2 x$$

Step-by-Step Solution

Answer

Period: \(\pi\), Amplitude: 1, Graph from 0 to \(2\pi\).

Step 1: Understand the Standard Form

The function given is of the form \(y = a \cdot \cos(bx + c) + d\). For this function, \(a = 1\), \(b = 2\), \(c = 0\), and \(d = 0\). The amplitude is given by \(|a|\), which is \(|1| = 1\), the coefficient of \(b\) determines the period.

Step 2: Calculate the Period

The period of a cosine function in the form \(y = \cos(bx)\) is given by \(\frac{2\pi}{b}\). Substitute \(b = 2\) to find the period: \(\frac{2\pi}{2} = \pi\). Thus, the period is \(\pi\).

Step 3: Graph Over Two Periods

Since the period is \(\pi\), two periods would be \(2\pi\). Therefore, we linearly plot the function from \(x = 0\) to \(x = 2\pi\). Start by plotting key points: at \(x = 0\), \(\pi\), and \(2\pi\), the function will have maximum values of 1, while at \(x = \frac{\pi}{2}\) and \(\frac{3\pi}{2}\), it will cross zero. Additionally, plot the minimum values of -1 at \(x = \frac{\pi}{4}\) and \(\frac{5\pi}{4}\), and mirror the pattern for the second period.

Step 4: Verify Amplitude and Period

Check the graph to ensure the height or amplitude is consistent, with peaks at 1 and troughs at -1. The period must repeat every \(\pi\) interval. Verify that the graph visually confirms having two full cycles from \(0\) to \(2\pi\).

Key Concepts

Understanding Amplitude

Amplitude is a crucial concept in trigonometric functions, particularly when dealing with sine and cosine functions. In general, the amplitude refers to the maximum height of the wave from its central axis or midline. It indicates how "tall" the wave is.

For the cosine function, given in the form \(y = a \cdot \cos(bx + c) + d\), the amplitude is determined by the absolute value of \(a\).

• If \(a = 1\), then the amplitude is simply 1, meaning the highest point of the wave is at 1, and the lowest is at -1.
• An amplitude of 1 indicates that no stretching or compressing occurs in the vertical direction.
  
  

This consistency in amplitude is useful for ensuring the accuracy of the graphical representation. Regardless of the phase shift or period change, the wave's amplitude remains dependent solely on the value of \(a\).

Exploring Periodicity

Periodicity in trigonometric functions refers to how often the function repeats itself over a specific interval. It's a vital characteristic that helps in predicting the behavior of the function.

When considering \(y = \cos(bx)\), the standard period of the cosine function is \(2\pi\). However, the period can change based on the coefficient \(b\).

• The formula to find the period is \(\frac{2\pi}{b}\).
• In the given function \(y = \cos(2x)\), substituting \(b=2\) yields a period of \(\pi\).

This means the function completes one full cycle every \(\pi\) units along the x-axis, doubling the frequency of the standard cycle. Understanding this helps in plotting the function correctly over specified intervals.

Features of the Cosine Function

Cosine is one of the fundamental trigonometric functions, often represented in the form \(y = \cos(x)\). Its graph is a wave-like pattern that oscillates between -1 and 1. Key features of the cosine function include:

• The range, which extends from -1 to 1, dictates the envelope within which the function oscillates.
• Unlike the sine function, the cosine function starts at its maximum value when \(x = 0\).
• It exhibits even symmetry and reflects symmetrically about the y-axis, centered at \(x = 0\).

Cosine waves occur naturally in various physical phenomena such as sound waves, light waves, and tides. The transformation parameters \(a\), \(b\), \(c\), and \(d\) in \(y = a \cdot \cos(bx + c) + d\) allow this function to model more complex periodic behaviors.