Problem 18 - Algebra and Trigonometry

Question

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: Identify the Function Form

The provided function is \( y = \frac{1}{2} \cos 4x \). This function is in the form \( y = a \cos(bx + c) + d \). Here, \( a = \frac{1}{2} \), \( b = 4 \), \( c = 0 \), and \( d = 0 \).

Step 2: Determine the Amplitude

The amplitude of a cosine function \( y = a \cos(bx) \) is given by the absolute value of \( a \). For this function, \( a = \frac{1}{2} \), so the amplitude is \( \left| \frac{1}{2} \right| = \frac{1}{2} \).

Step 3: Calculate the Period

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

Step 4: Sketch the Graph

To sketch the graph of \( y = \frac{1}{2} \cos 4x \), note the amplitude is \( \frac{1}{2} \), so the graph will oscillate between \(-\frac{1}{2}\) and \(\frac{1}{2}\). The period is \( \frac{\pi}{2} \), meaning one full cycle completes over this interval. The graph of cosine starts at its maximum, so begin at \( \frac{1}{2} \), go down to \( 0 \) at \( \frac{\pi}{4} \), and to \(-\frac{1}{2}\) at \( \frac{\pi}{2} \), then back symmetrically.

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry. It is often written as \( \cos(x) \), and represents the x-coordinate of a point on the unit circle. The general form of a cosine function is \( y = a \cos(bx + c) + d \). Each parameter in this equation affects the shape and position of the graph.

• Amplitude (\(a\)): This is the coefficient in front of the cosine function. It determines the height of the wave peaks and troughs. In the function \( y = \frac{1}{2} \cos(4x) \), the amplitude is \( \frac{1}{2} \), which indicates that the graph will oscillate between \(-\frac{1}{2}\) and \(\frac{1}{2}\).
• Period (\(b\)): This affects the frequency of the wave. The period is calculated by the formula \( \frac{2\pi}{b} \). For \( y = \frac{1}{2} \cos(4x) \), \(b = 4\), resulting in a period of \( \frac{\pi}{2} \), meaning that the function repeats every \( \frac{\pi}{2} \) units.
• Phase Shift (\(c\)): This shifts the graph horizontally. In this case, \( c = 0 \), so there is no horizontal shift.
• Vertical Shift (\(d\)): This moves the graph up or down. Here, \( d = 0 \), so there is no vertical shift.

Understanding these parameters is crucial when analyzing the behavior of the cosine function and predicting changes in its graph.

Graph Sketching

Graph sketching is an essential skill in mathematics that helps visualize functions. For trigonometric functions like the cosine function, the process involves several steps to ensure accuracy.To sketch the graph of a function like \( y = \frac{1}{2} \cos 4x \):

• Determine Key Points: Since the amplitude is \( \frac{1}{2} \), the peaks and troughs will occur at \( \frac{1}{2} \) and \(-\frac{1}{2}\), respectively. The graph will start at its maximum value of \( \frac{1}{2} \) when \( x = 0 \).
• Calculate the Period: As mentioned earlier, the period is \( \frac{\pi}{2} \). This means the entire wave from starting point, going down to the minimum, and back up completes in \( \frac{\pi}{2} \) units along the x-axis.
• Plot the Graph: Start by plotting the peak at \( (0, \frac{1}{2}) \), the zero point at \( (\frac{\pi}{4}, 0) \), and the trough at \( (\frac{\pi}{2}, -\frac{1}{2}) \). Continue this pattern to complete the sinusoidal wave.

This structured sketching approach helps to correctly visualize the function's behavior over its period.

Trigonometric Functions

Trigonometric functions, including sine and cosine, are periodic and are foundational to understanding various mathematical concepts. These functions are used often in calculus, physics, and engineering.Some key characteristics of trigonometric functions are:

• Periodic Nature: Trigonometric functions repeat their values in regular intervals. For instance, \( \cos(x) \) has a natural period of \( 2\pi \), meaning it completes one cycle every \( 2\pi \) units on the x-axis.
• Amplitude: This measures the maximum displacement from the midline of the wave, as seen in the cosine function \( y = a \cos(bx) \).
• Applications: These functions model waves, such as sound waves, and oscillations, such as pendulum swings, because of their repetitive nature.

By understanding these characteristics, we can apply trigonometric functions to solve real-world problems and make predictions by analyzing patterns in the graph.