Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=3 \cos \frac{\pi}{2} x $$

Step-by-Step Solution

Answer

Amplitude is 3, period is 4, and phase shift is 0.

Step 1: Identify Amplitude

The formula for a cosine function is generally given by \( y = A \cos(Bx - C) + D \). The amplitude \( A \) is the coefficient of the cosine function. In the given equation \( y = 3 \cos \left( \frac{\pi}{2} x \right) \), the amplitude is \( 3 \).

Step 2: Calculate Period

The period \( T \) of a cosine function can be found using the formula \( T = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the cosine function. Here, \( B = \frac{\pi}{2} \). Thus, the period is \( T = \frac{2\pi}{\frac{\pi}{2}} = 4 \).

Step 3: Determine Phase Shift

The phase shift of a cosine function is given by \( \frac{C}{B} \). Since \( C = 0 \) in this equation (there is no horizontal shift), the phase shift is \( \frac{0}{\frac{\pi}{2}} = 0 \).

Step 4: Sketch the Graph

To graph \( y = 3 \cos \left( \frac{\pi}{2} x \right) \), start by marking the amplitude as 3 units up (maximum) and 3 units down (minimum) from the midline (\( y = 0 \)). The period is 4, so complete a full wave from 0 to 4 along the x-axis. Since there is no phase shift, the graph starts at \( x=0 \) at its maximum point (3) and follows the cosine pattern. Mark additional cycles if needed, extending the pattern from -4 to 4 along the x-axis.

Key Concepts

Amplitude

Amplitude refers to the maximum height a wave reaches from its midline, which essentially is how tall or deep the wave gets. In the context of a cosine function like \( y = 3 \cos \left( \frac{\pi}{2} x \right) \), the amplitude is the number multiplying the cosine, which is 3 in this example. This means the wave will reach a maximum height of 3 units and a minimum of -3 units, centering around the horizontal midline, usually \( y = 0 \).

To understand amplitude in everyday terms, imagine it as the energy of the wave. More amplitude means more energy, making the wave visibly higher and deeper. For different functions:

• If the amplitude is larger, the wave stretches taller and wider from the axis.
• An amplitude of 1 implies the wave reaches exactly one unit up and down from the center.

In practice, no matter what function you're dealing with, identifying amplitude is straightforward: just look at the number multiplying the cosine or sine function.

Period of a Function

The period of a function defines how long it takes for the wave to repeat itself, start to finish. It's an important aspect of any wave-like function because it tells us how frequently the pattern repeats over the x-axis.

For a standard cosine function like \( y = A \cos(Bx - C) + D \), the period \( T \) can be calculated using the formula \( T = \frac{2\pi}{|B|} \). In our specific equation, \( y = 3 \cos \left( \frac{\pi}{2} x \right) \), the coefficient \( B \) is \( \frac{\pi}{2} \). Applying the period formula:

• \( T = \frac{2\pi}{\frac{\pi}{2}} = 4 \)

Thus, the period here is 4, meaning one full cycle of the pattern completes over a stretch of 4 units on the x-axis.

This cycle involves the graph going from a peak, through a trough, and back to another peak. Understanding and calculating the period helps in predicting how the function behaves as it continues along the axis.

Phase Shift

Phase shift in a cosine function determines where the wave pattern starts along the horizontal axis, essentially moving the wave left or right from its typical starting point without changing its shape or direction.

In the equation \( y = A \cos(Bx - C) + D \), the phase shift is calculated as \( \frac{C}{B} \). In our example, \( y = 3 \cos \left( \frac{\pi}{2} x \right) \), there is no \( C \) term, meaning \( C = 0 \). Hence the phase shift is \( \frac{0}{\frac{\pi}{2}} = 0 \).

This tells us the graph begins as a typical cosine graph would at \( x = 0 \), aligning perfectly without any horizontal translation.

• A positive phase shift might 'delay' the start of the wave, moving it to the right.
• A negative phase shift advances it to the left.

Understanding phase shift is crucial when dealing with complex functions, as it can greatly affect the visual representation and pattern placement of a function on a graph.