Problem 17 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(\pi\); participate in cycles between these values on the graph.

Step 1: Identify the Amplitude

The amplitude of a cosine function in the form \(y = a \cos(bx)\) is the absolute value of \(a\). In this equation, \(y = \cos(2x)\), the coefficient \(a\) is implicitly 1. Thus, the amplitude is \(|1| = 1\).

Step 2: Determine the Period

The period of a cosine function in the form \(y = a \cos(bx)\) is calculated using the formula \(\frac{2\pi}{b}\). Here, \(b = 2\). Hence, the period is \(\frac{2\pi}{2} = \pi\).

Step 3: Sketch the Graph

To sketch the graph of \(y = \cos(2x)\), note that the amplitude is 1, and the period is \(\pi\). This means that one full cycle of the cosine wave occurs between \(x = 0\) and \(x = \pi\). The typical points of the cosine function (maximum, zero, minimum, zero, maximum) should be plotted along one period and repeated in both positive and negative \(x\) directions. At \(x = 0\), \(y = 1\); at \(x = \frac{\pi}{2}\), \(y = 0\); and at \(x = \pi\), \(y = -1\). Repeat this pattern for additional cycles.

Key Concepts

The Cosine Function

The cosine function is a fundamental trigonometric function often used in mathematics to model periodic phenomena. It is denoted as \( \cos(x) \) and typically produces a wave-like graph. The cosine function is related to the unit circle, where it represents the horizontal coordinate of a point on the circle as the angle increases or decreases. It features prominently in various scientific and engineering fields.

Some characteristics of the cosine function include:

• The function is periodic, which means it repeats its values over regular intervals.
• It is an even function, implying that \( \cos(-x) = \cos(x) \), making its graph symmetric around the y-axis.
• Standardly oscillates between -1 and 1.

When applied to a function such as \( y = \cos(2x) \), it is subject to transformations that include changes in amplitude and period, both of which significantly affect the graph's shape and behavior.

Understanding Amplitude

Amplitude is a crucial concept when examining trigonometric functions. In simple terms, the amplitude of a function like \( y = a \cos(bx) \) measures the height of the wave from its middle value to its peak. This width is essential in determining how much the function will rise or fall from its central axis.

For \( y = a \cos(bx) \):

• The amplitude is \( |a| \).
• It determines the range of the function, showing a maximum of \( a \) and a minimum of \( -a \).

In the exercise \( y = \cos(2x) \), \( a \) equals 1, indicating an amplitude of 1. This means the graph ranges from 1 to -1, maintaining its standard height without vertical stretching or shrinking. Recognizing the amplitude helps in sketching and understanding the behavior of trigonometric graphs.

Period of Functions

The period of a trigonometric function tells how often the shape of the graph repeats. This concept is vital when analyzing functions because it allows us to predict and model cyclic behavior. To find the period of the function \( y = a \cos(bx) \), you can use the formula \( \frac{2\pi}{b} \).

Key points about function periods:

• A smaller period results in more frequent repetition of the wave pattern, leading to a more compressed graph.
• A larger period means the wave stretches further along the x-axis before repeating.

For \( y = \cos(2x) \), the coefficient \( b \) here is 2. Consequently, the period becomes \( \frac{2\pi}{2} = \pi \). This information means that every \( \pi \) units along the x-axis, the pattern of the cosine wave will start repeating. Understanding the period enhances your ability to sketch the graph correctly and comprehend how scaling impacts the waveform.