Problem 12 - Contemporary Precalculus

Question

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$4 / 5,3,1$$

Step-by-Step Solution

Answer

Answer: The rule of the function is \(y = \frac{4}{5}\sin\left(\frac{2\pi}{3}(x - 1)\right)\).

Step 1: Find the value of B using the Period

Use the formula \(B = \frac{2\pi}{P}\) to find the value of B, where \(P\) is the given period (3). $$B = \frac{2\pi}{3}$$

Step 2: Write down the rule of the function using the given amplitude, B, and phase shift

Now that we know the values of A, B, and C, plug them into the trigonometric function equation: $$y = A\sin(B(x - C)) + D$$ Using the given values: \(A = \frac{4}{5}\) \(B = \frac{2\pi}{3}\) \(C = 1\) \(D = 0\) We get the rule of the function as: $$y = \frac{4}{5}\sin\left(\frac{2\pi}{3}(x - 1)\right)$$

Key Concepts

Amplitude

Amplitude is a crucial aspect of periodic functions like sine and cosine functions. Think of the amplitude as the maximum height of a wave. It's the distance from the midline of the function to its peak or trough. In the context of a sine wave, the amplitude affects how tall or short the waves appear.

For example, if the amplitude is large, the wave peaks and troughs will be further from the midline, making the wave taller. If the amplitude is smaller, the wave will be shorter and closer to the midline.

• The amplitude is always a positive value.
• In the given exercise, the amplitude is \(\frac{4}{5}\).
• It arrests the vertical stretch or compression of the wave.

Without the right amplitude, a periodic function would lose its characteristic wave pattern.

Period

The period of a trigonometric function is the distance along the x-axis that it takes for the wave to start repeating itself. Essentially, it's how long it takes for the cycle of the wave to complete one full loop.

Mathematically, the formula to determine period \(B\) using \(P\) is \(B = \frac{2\pi}{P}\). For our exercise, the period \(P\) is 3, resulting in a \(B\) value of \(\frac{2\pi}{3}\).

• The period determines how "squished" or "stretched" the wave will be along the x-axis.
• Shorter periods result in waves closer together, whereas longer periods stretch them apart.

This concept helps us understand how frequently the waves repeat, which is pivotal in describing the behavior of periodic functions.

Phase Shift

Phase shift refers to the horizontal shift of a periodic function along the x-axis. It determines where the wave starts in its cycle. If the phase shift is positive, the wave is pushed to the right. Conversely, a negative phase shift moves the wave to the left.

In our exercise, the phase shift \(C\) is 1, indicating a rightward shift of one unit.

• The formula for incorporating phase shift in the function is \(y = A\sin(B(x - C)) + D\).
• Phase shifts adjust where the wave begins.
• A phase shift can change the point at which the function reaches its peak or zero.

Understanding phase shifts is crucial for accurately positioning the periodic function on a graph.

Trigonometric Function

A trigonometric function is a function related to the angles of triangles. The most common ones are sine, cosine, and tangent functions, which are essential for describing periodic phenomena.

In the problem presented, we focus on the sine function. Trigonometric functions are periodical, meaning they repeat their values in regular intervals.

• They can model waves, oscillations, and circular motion.
• Each function is characterized by its unique properties, like amplitude, period, and phase shift.
• They are widely used in physics, engineering, and signal processing.

In the sine function given: \(y = \frac{4}{5}\sin\left(\frac{2\pi}{3}(x - 1)\right)\), we see all these properties interacting together to describe the sine wave's behavior.