Problem 19 - Precalculus

Question

In Exercises \(11-20,\) state the amplitude and period of each function. $$y=5 \sin \left(\frac{\pi}{3} x\right)$$

Step-by-Step Solution

Answer

Amplitude: 5, Period: 6.

Step 1: Identify the standard form

The standard form of the sine function is \( y = a \sin(bx) \), where \( a \) represents the amplitude and \( b \) affects the period of the function.

Step 2: Identify the amplitude

In the function \( y = 5 \sin \left(\frac{\pi}{3} x\right) \), the coefficient \( a \) is 5. Therefore, the amplitude is 5.

Step 3: Determine the period using the formula

The period of a sine function \( y = \sin(bx) \) is given by the formula \( \frac{2\pi}{b} \). Identify \( b \) from the given function.

Step 4: Calculate the period

For the function \( y = 5 \sin \left(\frac{\pi}{3} x \right) \), \( b = \frac{\pi}{3} \). Thus, the period is \( \frac{2\pi}{\frac{\pi}{3}} = 6 \).

Key Concepts

Amplitude

The amplitude of a trigonometric function like the sine function represents how much the curve reaches above and below its central axis, essentially measuring the function's vertical stretch. For a sine function in the form \( y = a \sin(bx) \), the coefficient \( a \) defines the amplitude.

In the function \( y = 5 \sin \left(\frac{\pi}{3} x\right) \), the coefficient \( a \) is 5. This signifies that the maximum height of the wave from its central line is 5 units and the minimum depth is also 5 units below. Thus, the amplitude is consistent at 5 regardless of the transformation or stretch applied to the function. The amplitude is always a positive value, indicating the maximum extent of oscillation from the equilibrium line.

• Amplitude shows the height of the wave peak.
• In \( y = a \sin(bx) \), it's determined by \( |a| \).
• For this function, amplitude = 5.

Period of Sine Function

The period of a sine function describes how long it takes for the function to complete one full cycle of its pattern. It is the horizontal distance over which the sine wave repeats itself. For the sine function \( y = \sin(bx) \), the period is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) influences how quickly or slowly the cycle repeats.

To determine the period for the function \( y = 5 \sin \left(\frac{\pi}{3} x \right) \), identify \( b \) which is \( \frac{\pi}{3} \). Plugging into the formula gives the period as \( \frac{2\pi}{\frac{\pi}{3}} \). Simplifying this fraction results in 6, meaning that the wave pattern repeats itself every 6 units along the horizontal axis.

• The formula for the period is \( \frac{2\pi}{b} \).
• In this problem, \( b = \frac{\pi}{3} \).
• The calculated period is 6.

Standard Form of Sine Function

The standard form of a sine function is a mathematical expression that defines how the function is structured and transformed. It is represented by the formula \( y = a \sin(bx) \), where:

• \( a \) is the amplitude, giving the measure of how tall the wave reaches.
• \( b \) is a value that affects the period, determining how frequently the cycles occur over a set distance.

The parameters in this form allow for easy identification and calculation of crucial properties like amplitude and period.

Every alteration to these parameters modifies the graph's shape or size. For instance, increasing \( a \) will stretch the wave vertically, while tweaking \( b \) adjusts how condensed or elongated the waves are horizontally. By comparing the given function \( y = 5 \sin \left(\frac{\pi}{3} x \right) \) with the standard form, we can readily identify \( a \) and \( b \) to analyze the function further.

• The form is \( y = a \sin(bx) \).
• Helps easily determine amplitude via \( a \) and period via \( b \).
• Facilitates function analysis and transformation understanding.