Problem 20 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=5 \cos \frac{1}{4} x $$

Step-by-Step Solution

Answer

Amplitude = 5; Period = 8π.

Step 1: Identify the Standard Form

The standard form of a cosine function is given by \( y = a \cos(bx + c) + d \). In this exercise, we have \( y = 5 \cos \frac{1}{4} x \). We can identify the coefficients as follows: \( a = 5 \), \( b = \frac{1}{4} \), \( c = 0 \), and \( d = 0 \).

Step 2: Determine the Amplitude

The amplitude of a cosine function is the absolute value of \( a \), which represents the vertical stretch or compression of the graph. Here, \( a = 5 \), so the amplitude is \(|5| = 5\).

Step 3: Calculate the Period

The period of a cosine function is given by \( \frac{2\pi}{b} \). Substituting \( b = \frac{1}{4} \), we find the period as follows:\[\text{Period} = \frac{2\pi}{\frac{1}{4}} = 2\pi \times 4 = 8\pi.\]

Step 4: Sketch the Graph

To sketch the graph, note that the cosine function starts at its maximum when \( x = 0 \) and follows a typical cosine wave pattern. With an amplitude of 5, the graph will oscillate between 5 and -5. The period is \( 8\pi \), meaning one full cycle of the graph from maximum through minimum and back to maximum completes over an interval of \( 8\pi \) units on the x-axis. Plot key points every \( \frac{1}{4} \times 8\pi = 2\pi \) (i.e. at 0, \( 2\pi \), \( 4\pi \), ...) and use them to sketch the curve.

Key Concepts

Amplitude

The amplitude of a function refers to the height of its peaks or the depth of its troughs from the middle value, often known as the equilibrium position. For cosine functions, the amplitude determines how far the graph stretches or compresses vertically. In mathematical terms, it is represented as the absolute value of the coefficient `a` in the function's standard form:

• Standard Form: \( y = a \, \cos(bx + c) + d \)

This means if you see a function like \( y = 5 \, \cos \left( \frac{1}{4} x \right) \), the amplitude can be found by taking the absolute value of the coefficient `a`. Here, it is straightforward:

• Given \( a = 5 \), so the amplitude is \( |5| = 5 \).

In real-life terms, this value dictates that from the central axis, or middle value on the graph, the curve will reach as high as 5 units up and as low as 5 units down. Understanding amplitude is crucial as it helps you quickly determine how large the oscillation of the wave is.

Period

The period of a cosine function tells you the length of one complete cycle of the wave. This cycle includes going from peak to peak or from any starting point back to the same point after having gone through one full oscillation. For cosine functions, the standard way to figure out a period is to use the formula:

• \( \text{Period} = \frac{2\pi}{b} \)

In this exercise, we can see \( b = \frac{1}{4} \) in the function \( y = 5 \, \cos \left( \frac{1}{4} x \right) \). Plugging this value into the formula, you get:

• \( \text{Period} = \frac{2\pi}{\frac{1}{4}} = 2\pi \times 4 = 8\pi \).

So, the period of this function is \( 8\pi \). This means it takes \( 8\pi \) units along the x-axis for the function to complete one full cycle, so a point on the graph will reappear in the same position after moving along \( 8\pi \) units.

Cosine Function

The cosine function is a fundamental type of wave function often explored in trigonometry. Recognized by its unique wave-like appearance, the cosine function is periodic, meaning it repeats at regular intervals, defined by its period. The general form of a cosine function can be written as:

• \( y = a \, \cos(bx + c) + d \)

Each component of the formula modifies the wave's appearance:

• \( a \) affects the amplitude, dictating how tall or short the wave appears.
• \( b \) directly affects the period, indicating how long a full wave cycle takes.
• \( c \) shifts the wave left or right, known as the phase shift.
• \( d \) moves the wave up or down, adjusting the midline of the function.

In many practical applications, understanding how to adjust these parameters enables precise modeling of periodic phenomena, from waves in physics to cycles in economics. For \( y = 5 \, \cos \left( \frac{1}{4} x \right) \), we observe:

• The wave oscillates between 5 and -5 due to the amplitude.
• Completes one full cycle every \( 8\pi \) units due to the period.

The cosine function is a staple in trigonometry, aiding in the analysis and visualization of periodic behavior.