Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed in Problems 16-23. $$ y=3 \cos \frac{x}{2} $$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(4\pi\), no shifts.

Step 1: Identify the Standard Form

The function given is \( y = 3 \cos \left( \frac{x}{2} \right) \). The standard form for a cosine function is \( y = A \cos(Bx - C) + D \). In this case, \( A = 3 \), \( B = \frac{1}{2} \), \( C = 0 \), and \( D = 0 \).

Step 2: Determine the Amplitude

The amplitude of a cosine function \( y = A \cos(Bx - C) + D \) is given by the absolute value of \( A \). Here, \( A = 3 \), so the amplitude is \( 3 \).

Step 3: Calculate the Period

The period of a cosine function \( y = A \cos(Bx - C) + D \) is calculated as \( \frac{2\pi}{B} \). With \( B = \frac{1}{2} \), the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).

Step 4: Determine Horizontal and Vertical Shifts

Since \( C = 0 \) and \( D = 0 \), there are no horizontal or vertical shifts. The function \( y = 3 \cos \frac{x}{2} \) has its basic cosine shape centered at zero.

Step 5: Sketch the Graph

Given the amplitude \( 3 \) and period \( 4\pi \), the function \( y = 3 \cos \frac{x}{2} \) oscillates between \( -3 \) and \( 3 \). The interval \(-5 \leq x \leq 5\) is less than one full period, so the graph will rise and fall following the cosine curve starting from its maximum value at \( x = 0 \).

Key Concepts

Amplitude

The amplitude of a trigonometric function like the cosine function represents half the distance between its maximum and minimum values. It provides a measurement of the height of the oscillation above and below the centerline of the graph.
For the cosine function in standard form \( y = A \cos(Bx - C) + D \), the amplitude is represented by \( |A| \). In this solution example, \( y = 3 \cos \left( \frac{x}{2} \right) \), the amplitude is \( 3 \).
This means the graph oscillates between \( 3 \) and \( -3 \) along the y-axis.

• If \( A \) is positive, the graph will start at the maximum value, go down to the minimum, and return to maximum.
• If \( A \) were negative, the graph would start at its minimum value instead.

Understanding amplitude helps you predict how high or low the function will reach.

Period of a Function

The period of a trigonometric function is the horizontal length required for the function to complete one full cycle, returning to the same y-value at which it started. This measurement is critical for recognizing how frequently the function repeats itself over the x-axis.
For the general cosine function \( y = A \cos(Bx - C) + D \), the period is calculated using the formula \( \frac{2\pi}{B} \).
In the provided function, \( y = 3 \cos \left( \frac{x}{2} \right) \), \( B = \frac{1}{2} \). Thus, the period is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \).

• This means the function completes one cycle over \( 4\pi \) units.
• It's essential in graphing as knowing the period allows you to scale the x-axis properly.

Recognizing the correct period aids in predicting the timespan of repetition and helps sketch accurate graphs.

Cosine Function

The cosine function, one of the primary trigonometric functions, is defined in a repeating wave-like pattern. It's known for having its maximum value at the start of its cycle, which is why its graph starts at its peak.
The standard form is \( y = A \cos(Bx - C) + D \), where:

• \( A \) determines the amplitude, or vertical stretch of the graph.
• \( B \) affects the period, changing how quickly the wave repeats.
• \( C \) and \( D \) introduce horizontal and vertical shifts, respectively.

In our example, \( y = 3 \cos \left( \frac{x}{2} \right) \):

• The amplitude is 3, leading the function to range from -3 to 3.
• The period is \( 4\pi \), altering the speed of repetition.
• There are no shifts, keeping the graph centered on the y-axis.

The cosine function is essential in many fields, like physics and engineering, where wave motion is studied.

Graphing Trigonometric Functions

Graphing trigonometric functions is about showcasing how these mathematical expressions represent periodic phenomena. The challenge lies in accurately reflecting amplitudes, periods, and phase shifts. Here's the approach for graphing \( y = 3 \cos \left( \frac{x}{2} \right) \):

• Start by considering the amplitude of 3, which informs how high and low the graph will go on the y-axis.
• Next, consider the period of \( 4\pi \). Your graph should show one full cycle over this interval.
• Note that there are no phase shifts or vertical shifts in this function.
• Split the period into four equal parts to locate key points, like maxima, minima, and intersection with the x-axis. These are usually at \( 0, \pi, 2\pi, 3\pi \), and \( 4\pi \).

Once these elements are understood, plot them over the interval from -5 to 5. This helps you understand the nature of wave patterns and periodicity.