Problem 28 - Precalculus with Calculus Previews

Question

Find the amplitude and period of the given function. Sketch at least one cycle of the graph. $$ y=\frac{5}{2} \cos 4 x $$

Step-by-Step Solution

Answer

Amplitude: \(\frac{5}{2}\); Period: \(\frac{\pi}{2}\).

Step 1: Identify the Amplitude

The amplitude of a cosine function in the form \(y = a \cos(bx)\) is the absolute value of \(a\). For the given function, \(y = \frac{5}{2} \cos 4x\), the coefficient \(a\) is \(\frac{5}{2}\). Thus, the amplitude is \(\left| \frac{5}{2} \right| = \frac{5}{2}\).

Step 2: Determine the Period

The period of a cosine function \(y = a \cos(bx)\) is determined using the formula \(\frac{2\pi}{b}\). In this function, \(b\) is \(4\). Therefore, the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).

Step 3: Sketch One Cycle of the Graph

To sketch one cycle, start from \(x = 0\) and use the period \(\frac{\pi}{2}\). The cosine function completes one full cycle at \(x = \frac{\pi}{2}\). Plot the points using the amplitude and period: at \(x = 0\), \(y = \frac{5}{2}\); at \(x = \frac{\pi}{8}\), \(y = 0\); at \(x = \frac{\pi}{4}\), \(y = -\frac{5}{2}\); and back to \(y = 0\) at \(x = \frac{3\pi}{8}\), and \(y = \frac{5}{2}\) at \(x = \frac{\pi}{2}\). Connect these points smoothly for a cosine wave.

Key Concepts

Understanding Amplitude

When working with trigonometric functions like sine and cosine, amplitude is an essential concept to grasp. Amplitude refers to the height of the wave from its central axis. For a function in the form \( y = a \cos(bx) \), the amplitude is given by the absolute value of \( a \). It's that simple!

Here's how to think about it:

• Imagine a horizontal line cutting through the wave. The amplitude is the vertical distance from this line to the wave's highest point (peak) or lowest point (trough).
• This measurement is always positive because it's the "distance" rather than a specific point.
• In the equation \( y = \frac{5}{2} \cos 4x \), the amplitude is \( \left| \frac{5}{2} \right| = \frac{5}{2} \).

By knowing the amplitude, you understand how tall the wave is, which is crucial when you start graphing the function.

What is the Period?

The period of a trigonometric function measures how long it takes for the cycle of the wave to repeat itself. This tells you the "length" of one full wave cycle on the horizontal axis. For the cosine function represented as \( y = a \cos(bx) \), the period can be calculated using the formula \( \frac{2\pi}{b} \).

Here's how to determine the period step by step:

• Identify the coefficient \( b \) in the equation. Here, \( b = 4 \).
• Use the formula: Calculate \( \frac{2\pi}{b} \), which becomes \( \frac{2\pi}{4} \).
• Solve the expression, resulting in \( \frac{\pi}{2} \).

This result, \( \frac{\pi}{2} \), tells us that the wave repeats every \( \frac{\pi}{2} \) units along the x-axis. Knowing the period helps you sketch the graph correctly by showing where to expect the same pattern to recur.

The Cosine Function Explained

The cosine function is one of the fundamental trigonometric functions, often seen in the form \( y = a \cos(bx + c) + d \). When graphing \( y = \frac{5}{2} \cos 4x \), it's notable for its distinct wave-like pattern.

Here are some key points about the cosine function:

• The graph starts at its maximum value if there's no phase shift involved. For \( y = \frac{5}{2} \cos 4x \), it starts at \( \frac{5}{2} \) when \( x = 0 \).
• The function is periodic, repeating its pattern every period, which is \( \frac{\pi}{2} \) in this case.
• It oscillates symmetrically around its horizontal axis, known as the midline (which is at \( y = 0 \) for this function).
• The pattern of the cosine wave involves moving from the maximum to the midline, then to the minimum, back to the midline, and returning to the maximum.

Understanding the cosine function's behavior helps in accurately sketching the wave, predicting its behavior over different intervals, and analyzing its properties in various applications.