Problem 11 - Contemporary Precalculus

Question

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$3 / 4,2,0$$

Step-by-Step Solution

Answer

Answer: The rule for the given periodic function is $$ y(t) = \frac{3}{4} \cdot\sin(\pi t) $$.

Step 1: Identify amplitude, period, and phase shift

We are given the following values: Amplitude (A): $$3 / 4$$ Period (T): $$2$$ Phase shift (φ): $$0$$

Step 2: Write the general equation for a periodic function

The general equation for a periodic function is: $$ y(t) = A \cdot\sin(\frac{2\pi}{T}(t+\phi)) $$

Step 3: Substitute given values into the general equation

Now, we will substitute the given values into the equation: $$ y(t) = \frac{3}{4} \cdot\sin(\frac{2\pi}{2}(t+0)) $$

Step 4: Simplify the equation

Simplify the equation by multiplying the inside of the sine function: $$ y(t) = \frac{3}{4} \cdot\sin(\pi t) $$ This is the rule for the given periodic function.

Key Concepts

Amplitude

The amplitude of a periodic function refers to the maximum distance from the function's median line or equilibrium position to its peak. In simple terms, it indicates how tall or short the waves of the function are.
For a function given by the general formula \( y(t) = A \cdot \sin\left(\frac{2\pi}{T}(t+\phi)\right) \), the amplitude is represented by the constant \( A \). In our example, the amplitude is \( \frac{3}{4} \).

This means that the highest point of the wave is \( \frac{3}{4} \) units above the equilibrium position, and the lowest point is \( \frac{3}{4} \) units below.

• Magnitude is always a positive number.
• It affects the vertical stretch of the function's graph.
• Both sine and cosine functions share similar amplitude characteristics.

Understanding amplitude is crucial because it determines the extent of oscillation of the function. It tells us how intense or mild the oscillation is, impacting both visual representation and real-world applications, such as sound or light wave intensity.

Period

The period of a periodic function is the horizontal length required for the function to complete one full cycle or oscillation. Essentially, it tells us how wide the wave is across the horizontal axis.
In our function \( \frac{3}{4} \cdot \sin(\pi t) \), the period is determined by the term \( \frac{2\pi}{T} \) within the sine function. Here, the period \( T \) is given as 2.

The period dictates the repetition interval of the wave.

• Shorter periods result in waves that repeat more frequently.
• Longer periods produce waves that have more space between cycles.

The calculation of the period can be visualized as the distance along the horizontal axis before the function starts to repeat its pattern again. In practical contexts, understanding the period helps us to predict the timing of events, such as tidal cycles or the alternating current in electronics. A consistent period is a hallmark of any periodic function.

Phase Shift

A phase shift in a periodic function indicates a horizontal displacement of the entire wave from its standard position.
It essentially tells us how far the start of the wave has been moved to the left or right along the x-axis.
In the formula \( y(t) = A \cdot \sin\left(\frac{2\pi}{T}(t+\phi)\right) \):

• \( \phi \) represents the phase shift.

In this example, \( \phi \) is 0, meaning there is no horizontal displacement.

This means the wave starts at the origin without any shift. Phase shifts are crucial for aligning the starting points of different waves, which can be necessary in synchronizing signals or matching models to observed data.

• A positive phase shift moves the wave to the left.
• A negative phase shift displaces it to the right.

Understanding phase shifts allows us to modify the starting point of the wave, offering control over various applications, from engineering to physics. Adjusting the phase can be key in fields like signal processing, where precise timing is crucial.