Problem 22 - Precalculus Mathematics for Calculus

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: \(8\pi\).

Step 1: Identify the Amplitude

The amplitude of a cosine function in the form \( y = a \, ext{cos}(bx) \) is the absolute value of \( a \). Here, \( a = 5 \), so the amplitude is \( |5| = 5 \).

Step 2: Calculate the Period

The period of a cosine function is given by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the cosine function. Here, \( b = \frac{1}{4} \), so the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).

Step 3: Sketch the Graph

Start by plotting a normal cosine wave with amplitude 5 and period \( 8\pi \). The cosine wave starts at \( (0,5) \), reaches \((2\pi, -5)\), and completes one cycle at \( (8\pi, 5) \). The key points are due to the cosine wave pattern: maximum, zero, minimum, zero, maximum, repeating every \( 8\pi \). Remember to reflect about the x-axis, reaching maxima at \( y = 5 \) and minima at \( y = -5 \).

Key Concepts

Cosine Function

A cosine function is a type of periodic function, meaning it repeats its pattern at regular intervals. The general form of a cosine function is given by \( y = a \cos(bx + c) + d \), where:

• \( a \) is the amplitude, which affects the height of the wave.
• \( b \) affects the period or the length of one complete cycle of the wave.
• \( c \) is the phase shift, determining the horizontal displacement.
• \( d \) shifts the graph vertically.

In the specific function \( y = 5 \cos \frac{1}{4} x \), the amplitude is \( 5 \), and there are no phase shifts or vertical shifts. This function starts at its maximum point at \( x = 0 \), due to the nature of the cosine wave, which signifies peaks at \( y = a \). Understanding these parameters helps in predicting and drawing the cosine function accurately.

Graph Sketching

Sketching the graph of a trigonometric function involves knowing the shape and key points of the wave. For the function \( y = 5 \cos \frac{1}{4} x \), the process can be broken down into several steps:

• The wave begins at \( (0, 5) \), which is the maximum point.
• After a quarter period (\( 2\pi \)), it crosses the x-axis at \( (2\pi, 0) \), heading to the minimum.
• Halfway through the period (\( 4\pi \)), it reaches the minimum point at \( (4\pi, -5) \).
• Completing three-quarters of the period (\( 6\pi \)) sees it crossing the axis again.
• Finally, it returns to a maximum at the full period \( (8\pi, 5) \).

When sketching, ensure to correctly mark these key intervals to see the wave pattern clearly, making it easy to replicate for several cycles almost seamlessly. Such visualization enhances comprehension of the trigonometric function behavior.

Trigonometric Periodicity

One of the fundamental aspects of trigonometric functions like the cosine is periodicity. The periodicity of a function is how often it repeats itself. For a function in the form \( y = a \cos(bx) \), the period is calculated as \( \frac{2\pi}{b} \). In this case, the function \( y = 5 \cos \frac{1}{4} x \) has a coefficient \( b = \frac{1}{4} \), giving a period of \( 8\pi \).
This means that every \( 8\pi \) units along the x-axis, the function's pattern—comprising a maximum, a zero crossing, a minimum, and another zero crossing—repeats.
Knowing the period helps in understanding how stretched or compressed the function looks on the graph. A larger period value indicates a more stretched wave, translating into longer distances between peaks. Recognizing these patterns in trigonometric periodicity is key to predicting and interpreting graphs effectively.