Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-2 \sin (4 x+\pi)$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=-2 \sin (4x + \pi)\) is 2 and its period is \(\pi / 2\). Make sure to include two full periods when graphing the function.

Step 1: Identify the amplitude

The amplitude of the function is the absolute value of the coefficient in front of the sine function. In the given function, the coefficient is -2, therefore the amplitude is \(|-2|\), which equals to 2.

Step 2: Find the period

The period of a sine function is found by dividing \(2\pi\) by the coefficient of \(x\), denoted here as \(b\). In this case, \(b\) is 4, hence the period of the function is \(2\pi / 4 = \pi / 2\).

Step 3: Graphing the function

Use a graphing utility to plot the function. Make sure to include two full periods of the function.

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions used regularly in mathematics and various applied sciences. At its core, the sine function relates an angle in a right triangle to the ratio of the length of the opposite side to the hypotenuse. In more advanced terms, it's represented as a smooth, periodic wave that oscillates above and below an axis.
In its simplest form, the sine function is written as \( y = \sin(x) \). When this wave is graphed, it starts at the origin, rises to a peak, descends through the axis, dips to a trough, and then returns to the start, completing one cycle.

• Standard sine function: \( y = \sin(x) \)
• Pattern: Oscillating between -1 and 1
• Repeats every \(2\pi\) radians

By modifying this standard format, like multiplying by a coefficient or adjusting the angle, the sine function can be tailored to specific requirements. Such modifications affect its amplitude, period, and phase, allowing it to model real-world phenomena like sound waves and light.

Amplitude

The amplitude of a sine function describes its maximum extent from the central position (axis) to its peak or trough. It determines how tall or short the graph appears on the coordinate plane.
In the equation \( y = -2 \sin (4x + \pi) \), the amplitude is determined by the coefficient of the sine function, which in this case is -2. Although the minus sign indicates a reflection about the x-axis, the amplitude is considered as the absolute value:

• Amplitude = \(|-2| = 2\)

It means the wave reaches a height of 2 units above and below the central axis.
Knowing the amplitude is essential, especially if you're trying to understand the energy or intensity of repeating phenomena. For instance, in sound, a greater amplitude correlates with louder sounds.

Period of a Function

The period of a sine function refers to the length over which the function completes one full cycle of its wave pattern. It's a measure of how quickly the wave repeats itself over the x-axis.
For the sine function \( y = -2 \sin (4x + \pi) \), the period is calculated using the formula:\[ \text{Period} = \frac{2\pi}{b} \]where \( b \) is the coefficient inside the sine function next to \( x \). In this example, \( b = 4 \), so:

• Period = \( \frac{2\pi}{4} = \frac{\pi}{2}\)

When graphing, it's important to draw at least two full cycles to observe the function’s complete behavior, which assists in recognizing patterns and predicting future values. The concept of a period is widely used in various fields, from electronics to economics, illustrating the repetition of a time series data.