Problem 7 - Applied Calculus

Question

Sketch graphs of the functions. What are their amplitudes and periods? $$y=4 \cos 2 x$$

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(\pi\)

Step 1: Identify the general form of the cosine function

The general form of the cosine function is \( y = A \cos(Bx + C) + D \), where \( A \) is the amplitude, \( B \) affects the period and frequency, \( C \) is the phase shift, and \( D \) is the vertical shift. For the function \( y = 4 \cos 2x \), we can see that it follows this form with \( A = 4 \), \( B = 2 \), \( C = 0 \), and \( D = 0 \).

Step 2: Determine the amplitude

The amplitude of a cosine function is the absolute value of \( A \). Thus, for \( y = 4 \cos 2x \), the amplitude is \( |4| = 4 \). This indicates the maximum value the function reaches above and below its midline.

Step 3: Determine the period

The period of a cosine function is given by \( \frac{2\pi}{B} \). For \( y = 4 \cos 2x \), since \( B = 2 \), the period is \( \frac{2\pi}{2} = \pi \). This describes how long it takes for the function to complete one full cycle.

Step 4: Sketch the graph

To sketch the graph of \( y = 4 \cos 2x \), start with the baseline cosine curve and note that it will have a higher amplitude, reaching 4 and -4. The graph repeats every \( \pi \) units instead of every \( 2\pi \) due to the doubled frequency. There are no phase shifts or vertical translations, so the function oscillates symmetrically around \( y=0 \).

Key Concepts

Amplitude

In trigonometric functions, specifically the cosine function, the **amplitude** represents the height or the maximum deviation from the central axis (midline) of the graph. It shows how high and low the wave reaches as it oscillates. For the function given, this would be a positive number that multiplies the cosine function, indicating how far the peaks and troughs of the wave extend above and below the midline.

A standard cosine function is expressed in the general form \[ y = A \, \cos(Bx + C) + D \], where the amplitude is reflected by \( A \). The absolute value of \( A \) provides the amplitude. Thus, for the function \( y=4 \cos 2x \):

• Amplitude, \( A = 4 \)
• The midline is \( y = 0 \) since there is no vertical shift \( D \)

This means the graph will oscillate between the points \( y = 4 \) and \( y = -4 \), moving 4 units above and below the horizontal axis. Understanding amplitude is crucial for interpreting and sketching trigonometric functions as it defines the wave's "stretch" or height.

Period

The **period** of a trigonometric function is the length of the interval required for the function to complete one full cycle of its repeating pattern. This is a key concept to understand as it influences the frequency of the wave, which describes how "tight" or "spread out" the waves appear.

In the standard form of a cosine function \[ y = A \, \cos(Bx + C) + D \], the period is derived from \( B \), influencing how many cycles fit into a specific range. The formula for finding the period of a cosine function is \[ \frac{2\pi}{B} \]. For the function \( y=4 \cos 2x \):

• \( B = 2 \)
• Period = \( \frac{2\pi}{2} = \pi \)

This indicates that the graph of the function completes one cycle every \( \pi \) units along the x-axis. Hence, shorter periods correspond to more cycles over a given span, directly illustrating the function's frequency. Understanding the period helps accurately sketch and interpret must core features of trigonometric graphs.

Cosine Function

The **cosine function** is one of the fundamental trigonometric functions, often depicted as a wave-like structure. Its basic form is \[ y = \cos x \], which oscillates between -1 and 1, with a default amplitude and period when unaltered by coefficients or constants. This wave starts at its maximum value when \( x = 0 \) and exhibits symmetry about the y-axis.

When analyzing a cosine function like \[ y = A \cos(Bx + C) + D \], one can see how each parameter modifies the basic wave:

• \( A \): Controls the amplitude (height of the wave)
• \( B \): Alters the period (spacing between peaks)
• \( C \): Determines horizontal shift (moves the graph left or right)
• \( D \): Indicates vertical shift (moves graph up or down)

For the function provided, \( y=4 \cos 2x \), the significant points to note are:

• Amplitude of 4: Larger waves both above and below the midline.
• Period of \( \pi \): More frequent cycles for the same x-interval.
• No phase shift \( (C = 0) \): Peaks align with typical cosine points.
• No vertical shift \( (D = 0) \): Oscillates around \( y = 0 \).

These modifications enable a deeper understanding of how the standard cosine curve adapts based on different equation parameters, aiding in precise graph sketching and function interpretation.