Problem 10 - Precalculus with Limits

Question

Finding the Period and Amplitude, find the period and amplitude. $$ y=\frac{3}{2} \cos \frac{\pi x}{2} $$

Step-by-Step Solution

Answer

The amplitude of the function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\) is \(\frac{3}{2}\) and the period is 4.

Step 1: Determine Amplitude

The coefficient of the cosine function is the amplitude, which is the factor by which the function gets 'stretched' or 'compressed' vertically. In the given function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\), the coefficient of the cosine function is \(\frac{3}{2}\), therefore, the amplitude is \(\frac{3}{2}\).

Step 2: Determine Period

The factor that 'stretches' or 'compresses' the function horizontally is determined by 'B' in the equation \(y=A \cos (Bx)\). The period of the function is calculated as the ratio \( \frac{2\pi}{|B|}\). In the given function \(y=\frac{3}{2} \cos \frac{\pi x}{2}\), B equals \(\frac{\pi}{2}\). Therefore, the period is \(\frac{2\pi}{|\frac{\pi}{2}|}\) which simplifies to 4.

Key Concepts

Amplitude

Amplitude represents how far the graph of a trigonometric function reaches vertically, from its midline to its peak or trough. To understand this better, think of it as the measure of the wave's height. In mathematical terms, for a function like \[ y = A \cos(Bx) \] the amplitude is given by \( |A| \). This value tells us how much the cosine wave stretches or compresses vertically.

• If \( A = 1 \), the wave retains its original height.
• With \( A > 1 \), the wave stretches taller.
• When \( 0 < A < 1 \), it compresses shorter.
• An amplitude of \( A = 0 \) would mean the wave is flat.

In our exercise, the amplitude is \( \frac{3}{2} \). This means the cosine wave reaches a height of \( \frac{3}{2} \) units above and below the midline. It's crucial in determining the maximum and minimum values that the cosine function can achieve.

Period

The period of a trigonometric function tells us how long it takes for the wave to complete one full cycle and start repeating itself. Every trigonometric function has a specific period, and for the cosine function, this is influenced by the variable \( B \) in the equation: \[ y = A \cos(Bx) \]The formula to find the period \( T \) is: \[ T = \frac{2\pi}{|B|} \]This formula reflects how "squeezed" or "stretched" the wave will appear horizontally.

• If \( B = 1 \), the wave completes a cycle in the standard period of \( 2\pi \).
• With \( B > 1 \), the period decreases, making the wave complete more cycles in the same horizontal space.
• Conversely, when \( 0 < B < 1 \), the period lengthens and the wave takes longer to complete one cycle.

In our scenario, we have \( B = \frac{\pi}{2} \), so the period is calculated as \( \frac{2\pi}{\frac{\pi}{2}} = 4 \). This means the cosine function repeats its pattern every 4 units, making it more stretched out than the standard cosine wave.

Cosine Function

The cosine function is one of the fundamental trigonometric functions and is particularly known for its wave-like pattern. The basic form of the cosine function is given by \[ y = A \cos(Bx) \] where \( A \) and \( B \) play key roles in altering the wave's appearance. The cosine function is distinguished by its characteristic starts. It begins at its maximum value when at \( x = 0 \), unlike sine, which starts at zero.Key features of the cosine function include:

• Symmetry: The cosine curve is symmetrical about its vertical axis, meaning \( \cos(-x) = \cos(x) \).
• Range: The values of a basic cosine wave range from \(-1\) to \(1\). For our function \(y = \frac{3}{2} \cos \frac{\pi x}{2}\), this range scales to \(-\frac{3}{2}\) to \(\frac{3}{2}\).
• Periodicity: As mentioned, the cosine wave repeats every period \( T \), which is determined by \( B \). In our exercise, the period is 4.

Understanding these characteristics helps in graphing the function and predicting the wave's behavior over different domains. The cosine function is widely used in various fields like physics and engineering to describe periodic phenomena such as sound waves and light.