Problem 15 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=\cos 2 x $$

Step-by-Step Solution

Answer

Amplitude is 1; Period is \( \pi \).

Step 1: Identify the Amplitude

The general form of a cosine function is \( y = a \cos(bx + c) + d \), where \(a\) is the amplitude. In the given function \( y = \cos 2x \), \(a = 1\). Therefore, the amplitude is 1.

Step 2: Determine the Period

The period of a cosine function \( y = a \cos(bx + c) + d \) is given by the formula \( \frac{2\pi}{|b|} \). For the function \( y = \cos 2x \), \(b = 2\). So the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Sketch the Graph

To sketch the graph with the determined amplitude and period:1. Start by plotting the cosine graph which typically starts at \( (0, a) \) and goes through one cycle till \( (2\pi, a) \).2. For \( y = \cos 2x \), adjust the x-axis intervals to complete a cycle by \( \pi \).3. The graph will complete one full cycle between \( 0 \) and \( \pi \), with maximum at \( x = 0 \), minimum at \( x = \frac{\pi}{2} \), and again maximum at \( x = \pi \).4. The amplitude remains 1, so the peak is at 1 and the trough is at -1.

Key Concepts

Understanding Amplitude

When discussing trigonometric functions, the term "amplitude" is frequently mentioned. But what exactly is amplitude? Simply put, amplitude refers to the maximum height or distance from the average value to the peak of the wave.
In mathematical terms, it determines how far the peaks and troughs of a function are from the middle. For the function given, \( y = \cos 2x \), we observe that it follows the standard cosine function form, \( y = a \cos(bx + c) + d \). Here, the coefficient \( a \) before the cosine function is 1, which directly tells us the amplitude.
This means the graph will oscillate, or move up and down, from its midpoint to a maximum height of 1 and a minimum height of -1.

• The peak value—also called the crest—reaches 1.
• The lowest value—referred to as the trough—reaches -1.

Having a clear understanding of amplitude is crucial because it helps predict the vertical stretch or compression of the graph.

Determining the Period of a Function

The concept of the "period" of a trigonometric function is crucial in understanding its behavior over an interval. The period refers to the length of the smallest interval over which the function repeats itself.
In simpler words, it's how long it takes for the function to complete one full cycle. For any basic cosine or sine function \( y = a \cos(bx + c) + d \), the formula used to determine the period is \( \frac{2\pi}{|b|} \).
For our specific function \( y = \cos 2x \), \( b \) is 2. Therefore, the period is calculated as follows:

• Plug \( b = 2 \) into the formula \( \frac{2\pi}{|b|} \)
• This becomes \( \frac{2\pi}{2} = \pi \)
• The function completes one cycle every \( \pi \) units.

This means that every time you move \( \pi \) units along the x-axis, the cosine function will have completed one full oscillation. Understanding the period is essential for not just sketching graphs, but also in various applications such as signal processing.

Graphing Trigonometric Functions Simplified

Graphing trigonometric functions may initially seem challenging, but with a little breakdown, even the most complex graphs become manageable. As given in the exercise, the task is to sketch \( y = \cos 2x \), knowing its amplitude and period.
To start, remember that the typical cosine graph begins at its peak when \( x = 0 \):

• Since the amplitude is 1, the start point is \( (0, 1) \).
• The function will complete its cycle by \( x = \pi \).
• Halfway through the cycle, at \( x = \frac{\pi}{2} \), the graph hits its trough: \( y = -1 \).
• Finally, it returns to the peak at \( x = \pi \).

Follow these steps:
1. Plot the initial maximum point, \( (0, 1) \).
2. Move to the minimum at \( (\frac{\pi}{2}, -1) \).
3. Complete the cycle back at \( (\pi, 1) \).
This simple transformation of the basic cosine graph by compressing it horizontally to complete a cycle in half the usual time, due to the period \( \pi \), illustrates the essence of efficient trigonometric graphing.