Problem 28 - Algebra and Trigonometry

Question

\(17-28\) . Find the amplitude and period of the function, and sketch its graph. $$ y=-2+\cos 4 \pi x $$

Step-by-Step Solution

Answer

Amplitude is 1, period is \(\frac{1}{2}\). Graph oscillates around \(y = -2\) between \(-3\) and \(-1\).

Step 1: Identify the Function

The given function is \( y = -2 + \cos(4\pi x) \). This is a transformed cosine function.

Step 2: Determine the Amplitude

In the function \( y = -2 + \cos(4\pi x) \), the coefficient in front of the cosine function is 1 (since it's implied), so the amplitude is \( |1| = 1 \).

Step 3: Calculate the Period

The period of a cosine function \( y = \cos(bx) \) is given by \( \frac{2\pi}{b} \). Here, \( b = 4\pi \), so the period is \( \frac{2\pi}{4\pi} = \frac{1}{2} \).

Step 4: Sketch the Graph

To sketch the graph, start by noting the following key points: - The function oscillates around the line \( y = -2 \). - The amplitude of the oscillation is 1, meaning the maximum is \( -1 \) and the minimum is \( -3 \). - The period is \( \frac{1}{2} \), meaning it completes a full cycle every \( \frac{1}{2} \) units along the x-axis. - Label the x-axis from 0 to 1, ensuring \( x = 0 \) is a point of maximum at \( f(0) = -1 \), \( x = \frac{1}{4} \) a minimum point at \( f(\frac{1}{4}) = -3 \), and \( x = \frac{1}{2} \) returning to a maximum point at \( f(\frac{1}{2}) = -1 \).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function commonly used in mathematics and engineering. It describes the relationship between the angle and the adjacent side of a right triangle. In its most basic form, the cosine function is written as \( y = \cos(x) \). This function produces a wave-like pattern that oscillates between -1 and 1.

Key characteristics of the cosine function include:

• Oscillation: It oscillates between -1 and 1, creating a repeating wave pattern.
• Symmetry: The cosine function is an even function, meaning it is symmetric about the y-axis.
• Periodicity: It has a standard period of \( 2\pi \), which means the function repeats its values every \( 2\pi \) units.

In the exercise, the cosine function has been modified by an amplitude and period change, leading to the transformed function \( y = -2 + \cos(4\pi x) \). This transformation shifts the graph vertically and alters its wave length.

Amplitude Calculation

Amplitude refers to the measure of how much a trigonometric function, like cosine, diverges from its central axis to its extreme point. In simple terms, it represents the function's highest absolute point above or below its middle value.

To calculate the amplitude, identify the coefficient in front of the cosine term, represented here as \( |a| \). If the function does not explicitly show a coefficient, it is assumed to be 1. In this exercise, the function is \( y = -2 + \cos(4\pi x) \), where the coefficient is 1. Therefore, the amplitude is \( |1| = 1 \).

The amplitude impacts the range of the function by determining the maximum and minimum values. Here, the amplitude affects the graph by maintaining this range between one unit above and below its central axis, which is shifted downwards to \( y = -2 \). Thus, the maximum value is at \( y = -1 \) and the minimum is \( y = -3 \).

Period Calculation

The period of a trigonometric function defines the length of the x-axis over which the function completes one full cycle or oscillation. In mathematics, the standard period of the cosine function \( y = \cos(bx) \) is calculated using the formula \( \frac{2\pi}{b} \). The variable \( b \) represents the frequency factor that modifies how quickly the oscillations occur.

In the exercise \( y = -2 + \cos(4\pi x) \), the cosine term is \( 4\pi x \), meaning \( b = 4\pi \). By substituting \( b \) into the period formula, we calculate:

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

This calculation shows that each complete oscillation of the function occurs over \( \frac{1}{2} \) units on the x-axis. Therefore, the frequency of oscillations is much higher compared to the basic cosine function, compressing several cycles into a shorter interval.

Graph Sketching for Trigonometric Functions

Sketching the graph of a transformed trigonometric function requires understanding of all modifications made to the original function. With \( y = -2 + \cos(4\pi x) \), several key details help in drawing the graph accurately.

• Vertical Shift: Begin by recognizing the -2 outside the cosine function. This shifts the entire wave down by 2 units.
• Amplitude: Knowing that the amplitude is 1, the wave oscillates up to 1 unit above and below the line \( y = -2 \).
• Period: With a period of \( \frac{1}{2} \), the waveform completes a full cycle over each 0.5 units along the x-axis.

Using these steps:

• The curve should oscillate between \( y = -3 \) and \( y = -1 \).
• Mark key points such as \( x = 0 \) (maximum), \( x = \frac{1}{4} \) (minimum), and \( x = \frac{1}{2} \) (maximum again).
• Extend this pattern repetitively as needed to fit the desired range on the graph.

By following these structured guidelines, one can genuinely visualize how the transformations affect the overall shape of the cosine function, ensuring an accurate representation of the equation provided.