Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over the interval \([-2 \pi, 2 \pi] .\) Give the amplitude. $$y=2 \cos x$$

Step-by-Step Solution

Answer

The amplitude is 2.

Step 1: Identify the function to graph

We are given the function \( y = 2 \cos x \). This is a cosine function with an amplitude adjustment.

Step 2: Determine the amplitude

The amplitude of the cosine function \( y = A \cos x \) is given by the absolute value of \( A \). Here, \( A = 2 \), so the amplitude is \( |2| = 2 \).

Step 3: Identify the period

For the cosine function \( y = \cos x \), the period is \( 2\pi \). Since there is no horizontal stretching or compression, the period remains \( 2\pi \).

Step 4: Choose key points for sketching the graph

The key points for \( y = \cos x \) over one period are at \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \text{and } 2\pi \). Each of these points shifts the graph by a factor of the amplitude, \( 2 \).

Step 5: Apply the transformation for the amplitude

Multiply the typical values of \( \cos x \) (which are \( 1, 0, -1, 0, 1 \)) by \( 2 \). This gives the values for \( y = 2 \cos x \) as \( 2, 0, -2, 0, 2 \).

Step 6: Extend the pattern across the interval \([-2\pi, 2\pi]\)

Repeat the pattern of points from Step 4 symmetrically over the interval \([-2\pi, 2\pi]\). The graph mirrors itself across the y-axis, showing periodic behavior.

Step 7: Sketch the graph

Plot the points derived in Steps 5 and 6 on a graph and connect them smoothly with a cosine wave, emphasizing the maximum and minimum values at \( 2 \) and \( -2 \), respectively.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that is essential for graphing. It is denoted as \( y = \cos x \) and characterizes the x-coordinate of a point on the unit circle as it moves counterclockwise around the circle. This function varies between 1 and -1 as it smoothly oscillates, capturing the wave-like behavior.
Characteristics of the cosine function include:

• It is an even function, meaning \( \cos(-x) = \cos(x) \). This symmetry results in a graph that is mirrored across the y-axis.
• The cosine wave repeats every \( 2\pi \) units along the x-axis. This property is known as periodicity, critical for understanding the repetition in patterns.
• The standard form \( y = \cos x \) touches the maximum at 1, minimum at -1, and crosses the x-axis at \( \pi/2 \), \( 3\pi/2 \), where the value of \( \cos x \) is zero.

This simple oscillation becomes the foundation when adjusting with parameters like amplitude and period, allowing transformations like \( y = 2 \cos x \).

Amplitude

The amplitude in a cosine function like \( y = A \cos x \) is a crucial concept as it determines the height of the wave peaks and the depth of the troughs. Amplitude is derived from the coefficient \( A \) in the equation.

• In \( y = 2 \cos x \), the amplitude is 2. This means the graph stretches vertically so the maximum y-value is 2, and the minimum is -2.
• Amplitude represents the absolute maximum displacement from the equilibrium position, which would be the x-axis in a standard cosine graph.

The role of amplitude is like stretching or compressing a wave vertically. In practical applications, a higher amplitude might mean a louder sound or more intense light wave.
For any cosine function, the amplitude is always positive and is calculated as \( |A| \). Understanding this helps in modifying the basic cosine graph effectively.

Period of Functions

A period in trigonometric functions refers to the interval length over which the function's graph completes one full cycle. The period is an essential factor when graphing or understanding periodic behavior in functions.
The basic cosine function \( y = \cos x \) has a period of \( 2\pi \), meaning every \( 2\pi \) units along the x-axis, the wave starts repeating:

• A full cycle begins at any peak, descends to a trough, and ascends back to another peak.
• This property leads to periodic phenomena, where the function values repeat predictably.

When there are no horizontal stretches or compressions, the period remains the unchanged \( 2\pi \). However, horizontal transformations affect the period by altering the stretch:

• If the function is modified as \( y = \cos(bx) \), the period changes to \( \frac{2\pi}{|b|} \), reflecting how fast the wave cycles through a complete pattern.

Understanding periods allows us to predict and analyze cycles efficiently in various practical contexts like signal processing or seasonal changes.