Problem 9 - Contemporary Precalculus

Question

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$3, \pi / 4, \pi / 5$$

Step-by-Step Solution

Answer

Answer: The two possible rules for the given periodic function are $$y = 3\sin(8(x - \pi / 5))$$ and $$y = 3\cos(8(x - \pi / 5))$$.

Step 1: Identify the given values

We are given the amplitude, period, and phase shift of the periodic function: $$A = 3, \text{ Period }= \pi / 4, \text{ Phase Shift }= \pi / 5$$

Step 2: Find the value of B

The period (T) of a periodic function is related to B by the formula: $$T = \frac{2\pi}{|B|}$$ Here, we're given \(T=\pi / 4\). We can solve for B as follows: $$\pi / 4 = \frac{2\pi}{|B|}$$ $$|B| = \frac{2\pi}{\pi / 4}$$ $$|B| = 8$$ Since B can be positive or negative, we can have two possible values for B: B = 8 or B = -8.

Step 3: Write the rule of the periodic function using sine and cosine functions

Now, let's write the rule of the periodic function using sine and cosine functions with the given values of A, B, and C. For sine function, the rule is $$y = 3\sin(8(x - \pi / 5))$$ For cosine function, the rule is $$y = 3\cos(8(x - \pi / 5))$$ These are the two possible rules for the given periodic function using sine and cosine functions. Note that either function could represent the periodic function, as the only difference between them is the initial phase.

Key Concepts

Amplitude

Amplitude measures the height of a wave in a periodic function. Specifically, it indicates the wave's maximum displacement from its equilibrium position. In simpler terms, it's how tall the wave is.

• In the case of a sine or cosine function with an amplitude of 3, the wave will reach as high as 3 and as low as -3 from its midpoint.
• An amplitude of 3 means every peak and trough of the wave is 3 units away from the centerline.

The amplitude is crucial because it influences the energy and intensity represented by the wave. A higher amplitude means the wave is more pronounced, whereas a smaller amplitude indicates a less vigorous wave.

Period

The period of a function is the distance it takes for a wave to repeat itself. Essentially, it's the length of one complete cycle. For a sine or cosine function, the period is determined using the formula: \[ T = \frac{2\pi}{|B|} \]In our exercise, the period is given as \(\pi/4\). This means each full wave cycle completes over an interval of \(\pi/4\).

• This is a fraction of the usual period of a sine or cosine wave, \(2\pi\), meaning the wave repeats quickly.
• Smaller periods result in more cycles within a given range, creating a more oscillatory signal.

Understanding the period is key for predicting when a function returns to its starting value.

Phase Shift

Phase shift refers to the horizontal shift left or right of a wave function relative to its usual starting point. For the functions in our solution, the phase shift is \(\pi/5\).

• A phase shift of \(\pi/5\) to the right implies the entire wave is displaced \(\pi/5\) units in the positive x-direction.
• This modification does not affect the shape of the wave; it simply moves it along the x-axis.

Phase shifts are important when aligning different waves or signals, making them crucial in applications like sound engineering and telecommunications.

Sine Function

The sine function is one of the basic trigonometric functions and is used to describe smooth periodic oscillations. The general form of the sine function is:\[ y = A\sin(B(x - C)) \]where:

• \(A\) represents the amplitude.
• \(B\) determines the period.
• \(C\) is the phase shift.

In our solution, we use the sine rule: \( y = 3\sin(8(x - \pi/5)) \).

• This equation describes a wave with an amplitude of 3.
• A period influenced by \(B = 8\), resulting in quick oscillations.
• Shifted to the right by \(\pi/5\).

Applications of sine functions include modeling cyclical behaviors, such as tides, alternating current, and in music.

Cosine Function

Similar to the sine function, the cosine function describes periodic oscillations but starts differently on the graph. The general form of the cosine function is:\[ y = A\cos(B(x - C)) \]Here, it represents:

• \(A\), the amplitude.
• \(B\) affecting the period.
• \(C\) as the phase shift.

In our task, using the cosine rule: \( y = 3\cos(8(x - \pi/5)) \),

• The amplitude is 3.
• The wave oscillates rapidly due to \(B = 8\).
• There's a rightward shift by \(\pi/5\).

Cosine functions are widely used in electromagnetic waves, sound waves, and to model natural periodic phenomena.