Problem 22 - Algebra and Trigonometry

Question

\(17-28\) . 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 is 5, period is \(8\pi\).

Step 1: Identify the general form

The given function is \( y = 5 \cos \frac{1}{4} x \). The general form of a cosine function is \( y = A \cos(Bx + C) + D \). In this problem: \( A = 5 \), \( B = \frac{1}{4} \), \( C = 0 \), 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 \( A \). For this function, the amplitude \( |A| = |5| = 5 \).

Step 3: Find the period using B

The period of a cosine function is calculated by \( \frac{2\pi}{|B|} \). For this problem, \( B = \frac{1}{4} \), so the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).

Step 4: Identify phase shift and vertical shift

There are no phase shifts or vertical shifts in this function since \( C = 0 \) and \( D = 0 \). The graph will start at its maximum at \( x = 0 \).

Step 5: Sketch the graph

To sketch the graph, plot the cosine wave starting at its maximum value (5), reaching its minimum at half the period \( 4\pi \), and completing a full cycle at \( 8\pi \). The graph oscillates between 5 and -5 over each period of \( 8\pi \) with no shifts.

Key Concepts

Amplitude

When dealing with trigonometric functions like the cosine function, amplitude is a key concept. The amplitude represents the maximum value the function reaches above or below its central axis (usually the x-axis).
For a cosine function of the form:
\[ y = A \cos(Bx + C) + D \]The amplitude is given by \(|A|\).

• In our example, the function is \( y = 5 \cos \frac{1}{4}x \), making \(A = 5\).
• The amplitude is \(|5| = 5\).

This tells us the cosine wave will reach a peak of 5 units above its midline and drop to a minimum of 5 units below. It implies the total vertical distance covered by the wave from peak to trough is 10 units.
Understanding amplitude helps us visualize the strength or height of the wave.

Period

Period is another important aspect of graphing the cosine function. It refers to the horizontal length needed for the function to complete one full cycle.
To find the period for the standard form:
\[ y = A \cos(Bx + C) + D \]The formula used is \( \frac{2\pi}{|B|} \).

• In our example, \( B = \frac{1}{4} \).
• So, the period calculates as \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).

This period tells us that the cosine wave pattern repeats every \(8\pi\) units along the x-axis.
This means the graph will make a full swing from its maximum, through its minimum, and back to its maximum in that distance.

Graphing Trigonometric Functions

Graphing trigonometric functions, such as cosine, involves understanding their amplitude and period. These define the shape and extent of the wave.
For a function like \( y = 5 \cos \frac{1}{4}x \):

• The amplitude tells us the wave varies between 5 and -5.
• The period of \(8\pi\) defines the horizontal stretch or compression.

To draw this graph:

• Start at its maximum value of 5 at \(x=0\).
• As you move to the right, the graph descends until it reaches -5 at \(x=4\pi\), halfway through the period.
• Continue plotting by rising back to peak at \(x=8\pi\), completing one full cycle.

There are no shifts here since \(C = 0\) and \(D = 0\), simplifying the graphing process.
Understanding these basic graphing principles makes it easier to recreate and predict the behavior of the cosine wave.