Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=4 \sin \left(\frac{1}{3} x-\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(6\pi\), Phase Shift: \(-\pi\).

Step 1: Identify the Amplitude

The amplitude of a sine function is the absolute value of the coefficient in front of the sine function. Here, the coefficient is 4, so the amplitude is \( |4| = 4 \).

Step 2: Determine the Period

The period of a sine function is given by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the sine function. Here, \( b = \frac{1}{3} \), so the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).

Step 3: Calculate the Phase Shift

The phase shift is calculated by solving \( \frac{c}{b} \), where \( c \) is the constant term inside the parentheses, and \( b \) is the coefficient of \( x \). Here, \( c = \frac{\pi}{3} \) and \( b = \frac{1}{3} \), so the phase shift is \( -\frac{\pi/3}{1/3} = -\pi \), meaning the graph is shifted to the right by \( \pi \) units.

Step 4: Sketch the Graph

To sketch the graph, start by drawing the basic sine curve amplified by 4 over one period \( [0, 6\pi] \). Shift this graph to the right by \( \pi \) units to incorporate the phase shift. Mark the key points of the graph where the sine function reaches its maximum, minimum, and zero values in this range.

Key Concepts

Amplitude

Amplitude is one of the fundamental characteristics of a sine function. It measures the maximum extent of the wave from its equilibrium position (middle line) on the graph.
In simpler terms, imagine you are watching the waves at the beach. The amplitude is like the height of a wave from the calm sea level to its peak. For the function given:

• The amplitude is determined by the coefficient in front of the sine function. In the equation provided, this coefficient is 4.
• Therefore, the amplitude is simply 4, indicating the highest and lowest points on the graph will be 4 units above and below the center line.

Understanding amplitude helps us visualize how "tall" or "short" the waves of a sine function will appear relative to the neutral center line, which in this case is y = 0.

Period of a Function

The period of a function describes how often the function repeats its values. For sine functions, this involves one complete cycle of the wave pattern, from start to finish.
To determine the period of a sine function, one can use the formula:\[Period = \frac{2\pi}{b}\]

• Where \( b \) is the coefficient of \( x \) within the trigonometric function.
• In our equation, \( b = \frac{1}{3} \).
• Thus, the period is \( \frac{2\pi}{\frac{1}{3}} = 6\pi \).

A period of \( 6\pi \) means the sine wave takes this length on the x-axis to complete one full cycle. This understanding helps in graphing sine waves, ensuring you place the wave peaks and troughs at correct intervals.

Phase Shift

Phase shift refers to the horizontal translation of the graph of a sinusoidal function along the x-axis.
For sine functions, phase shift is calculated by:\[Phase\ Shift = -\frac{c}{b}\]Here:

• \( c \) is the constant term added or subtracted in the function's argument.
• \( b \) is the coefficient of \( x \) as seen in the sine function.
• In our exercise, \( c = \frac{\pi}{3} \) and \( b = \frac{1}{3} \).
• Thus, the phase shift is \(-\frac{\pi}{3} \cdot \frac{3}{1} = -\pi\).

This signifies a rightward shift of \( \pi \) units. Recognizing phase shifts allows you to properly position the start of the sine wave, so the graph accurately reflects this translation.

Sine Function

The sine function is a foundational trigonometric function that creates smooth, periodic wave patterns and is widely used in mathematics.
Its general form is:\[y = a \sin (bx + c)\]Here:

• \( a \) is the amplitude, affecting wave height.
• \( b \) determines the period of the wave.
• \( c \) leads to a phase shift.

The beauty of the sine function lies in its consistent oscillations, which model natural phenomena such as sound, light waves, and even tides. Understanding how to manipulate its variables lets us graph these waves accurately.In our specific example, the function is \( y = 4 \sin \left( \frac{1}{3}x - \frac{\pi}{3} \right) \), illustrating:

• A wave that reaches up to 4 units in height and depth.
• Repeats once every \( 6\pi \) units along the x-axis.
• Starts a cycle with a horizontal shift of \( \pi \) to the right.

Mastering these elements helps decode complex functions into manageable parts.