Problem 20 - Algebra and Trigonometry

Question

\(17-28\) . Find the amplitude and period of the function, and sketch its graph. $$ y=\frac{1}{2} \cos 4 x $$

Step-by-Step Solution

Answer

Amplitude = \(\frac{1}{2}\); Period = \(\frac{\pi}{2}\).

Step 1: Understanding the Function

The given function is \( y = \frac{1}{2} \cos 4x \). This is a type of cosine function with modifications in amplitude and frequency.

Step 2: Determine the Amplitude

The general form of a cosine function is \( y = A \cos(Bx) \), where \( A \) is the amplitude. Here, \( A = \frac{1}{2} \), so the amplitude of the function is \( \frac{1}{2} \).

Step 3: Calculate the Period

The period of a cosine function \( y = A \cos(Bx) \) is given by \( \frac{2\pi}{B} \). For our function, \( B = 4 \), thus the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \).

Step 4: Sketch the Graph

To sketch the graph, plot a cosine wave with an amplitude of \( \frac{1}{2} \) and a period of \( \frac{\pi}{2} \). This means every complete cycle of the wave spans an interval of \( \frac{\pi}{2} \) on the x-axis.

Key Concepts

Amplitude

The concept of amplitude in trigonometric functions, like the cosine function, refers to the height of the wave from the center line to the peak or trough. This can be visualized as the maximum vertical distance the graph covers. In general, the amplitude is a positive constant that stretches or compresses the graph vertically.

For the specific function given by \( y = \frac{1}{2} \cos 4x \), the amplitude is determined by the coefficient in front of the cosine function. Here, it is \( \frac{1}{2} \). This means that the highest point on the graph reaches \( \frac{1}{2} \) above the center line \( y = 0 \), and the lowest point reaches \( -\frac{1}{2} \).

To comprehend the impact of the amplitude, consider:

• A larger amplitude results in a taller graph.
• A smaller amplitude leads to a shorter graph.
• The amplitude is always expressed as a positive value, regardless of any negative signs in front of it.

Period

The period of a cosine function defines the length of one complete cycle of the wave on the x-axis. It tells us how quickly the wave repeats itself. In general, the period is influenced by the frequency component, represented by \( B \) in the function \( y = A \cos(Bx) \).

The formula to find the period of a cosine function is \( \frac{2\pi}{B} \). For \( y = \frac{1}{2} \cos 4x \), this gives us a period of \( \frac{2\pi}{4} = \frac{\pi}{2} \). This means it takes \( \frac{\pi}{2} \) units on the x-axis for the cosine wave to complete one full cycle.

Key points about the period include:

• A smaller period means the function completes its cycle more quickly, leading to more cycles over a given interval.
• A larger period indicates a slower cycle completion with fewer cycles over the same interval.
• The period is always measured in the x-axis units, such as radians or degrees.

Graph Sketching

Graph sketching involves plotting the function's main characteristics based on its amplitude, period, and other transformations. This creates a visual representation to understand the behavior of the function.

For the function \( y = \frac{1}{2} \cos 4x \), the process to sketch the graph involves:

• Identifying the amplitude as \( \frac{1}{2} \), which sets the maximum and minimum y-values.
• Calculating the period as \( \frac{\pi}{2} \), determining the interval for a full cycle.
• Plotting the key points: start at \( x = 0 \) with maximum point \( y = \frac{1}{2} \), reach the center line at \( x = \frac{\pi}{8} \), minimum point \( y = -\frac{1}{2} \) at \( x = \frac{\pi}{4} \), and complete the first cycle at \( x = \frac{\pi}{2} \).
• Continuing the wave pattern both to the left and right, maintaining the same amplitude and period in each cycle.

Graph sketching aids in understanding how the function behaves and highlights any changes in the graph due to adjustments in amplitude or period.