Problem 10 - Contemporary Precalculus

Question

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$4,5,0$$

Step-by-Step Solution

Answer

Answer: The rule for the periodic function is $$f(x) = 4\sin\left(\frac{2\pi}{5}x\right)$$.

Step 1: Determine the amplitude

The given amplitude is 4. This will be the value to scale the sine or cosine function. The amplitude will be placed as the coefficient of the sine or cosine function.

Step 2: Determine the period

The given period is 5. This is the time it takes for the function to complete one full cycle. To find the angular frequency ($$\omega$$) of the function, use the formula: $$\omega = \frac{2\pi}{T}$$ where T is the period. In this case, the period is 5, so the angular frequency will be: $$\omega = \frac{2\pi}{5}$$

Step 3: Determine the phase shift

The given phase shift is 0. This means that the function starts at the normal position and doesn't require any horizontal translation. Therefore, the phase shift can be ignored in this case.

Step 4: Write the rule of the periodic function

Using the amplitude, angular frequency, and phase shift, the rule of the periodic function can be written in the form of either sine or cosine function. In this case, let's write the rule using sine function: $$f(x) = A\sin(\omega x + \phi)$$ where A is the amplitude, $$\omega$$ is the angular frequency, and $$\phi$$ is the phase shift. Substitute the given values into the equation: $$f(x) = 4\sin\left(\frac{2\pi}{5}x + 0\right)$$ Simplify to get the final rule of the periodic function: $$f(x) = 4\sin\left(\frac{2\pi}{5}x\right)$$

Key Concepts

Amplitude

Amplitude refers to the height of a wave in a periodic function from its centerline to its peak or trough. It determines how far the wave's points are from its average position.
The amplitude of a sine wave affects its overall size and energy level. In the given function,

• The amplitude is 4.
• This means the highest and lowest points of the sine wave are 4 units above and below the centerline, respectively.

Amplitude doesn't affect the wave's timing or frequency, just its height. As such, when you see the number in front of the sine function, that's typically its amplitude.

Period

The period of a periodic function is the amount of time it takes to complete one full cycle. For sine waves, this is the length of one complete arc from start to finish.
You can calculate the period (\( T \)) with the formula:

• \[ T = \frac{2\pi}{\omega} \]

Here,

• The given period is 5.
• To match this with the sine function, we find the angular frequency, \( \omega \), using the formula we just mentioned, which gives us \( \omega = \frac{2\pi}{5} \).

What this tells you is that the function repeats itself every 5 units along the x-axis.

Phase Shift

Phase shift in a sine function determines how the wave is positioned horizontally. It tells us where the wave starts relative to the origin.
When constructing a sine function,

• A positive phase shift moves the graph to the right.
• A negative phase shift translates it to the left.
• A zero phase shift means the graph starts its cycle without horizontal displacement.

In the provided function, the phase shift is 0. Therefore, the sine wave begins right at the standard starting point of \( x = 0 \).

Sine Function

The sine function is one of the fundamental periodic functions in mathematics, often used to describe wave-like phenomena.

• It appears in the form \( \sin(x) \),
• and it oscillates between -1 and 1.

In this problem, you're given a combined sine function: \( f(x) = 4\sin\left(\frac{2\pi}{5}x\right) \).

• The number 4 represents the amplitude.
• The expression \( \frac{2\pi}{5} \) is part of its frequency and determines its speed.

Because of these characteristics, the sine function is central in describing periodic cycles like sound waves or the motion of pendulums.