Problem 18 - Precalculus

Question

In Exercises \(11-20,\) state the amplitude and period of each function. $$y=-2 \sin (\pi x)$$

Step-by-Step Solution

Answer

Amplitude: 2; Period: 2.

Step 1: Identify the Amplitude

The amplitude of a sine function in the form \(y = a \sin(bx)\) is given by the absolute value of the coefficient \(a\). Here, \(a = -2\). Therefore, the amplitude is \(|-2| = 2\).

Step 2: Determine the Period

The period of a sine function \(y = a \sin(bx)\) is given by \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the sine function. Here, \(b = \pi\). Therefore, the period is \(\frac{2\pi}{\pi} = 2\).

Key Concepts

Understanding the Sine Function

The sine function, represented as \( \sin(x) \), is a wave-like periodic function used to model oscillating behaviors. It is continuous, smooth, and repeats its values in a regular pattern.

The typical form of the sine function is \( y = a \sin(bx + c) + d \), where:

• \( a \) determines the amplitude.
• \( b \) affects the period.
• \( c \) influences the horizontal shift (or phase shift).
• \( d \) accounts for the vertical shift.

The basic sine function \( \sin(x) \) oscillates between \(-1\) and \(1\). This is why the amplitude, which we will discuss next, is so crucial. Also, knowing how these parameters alter the sine wave can assist in visualizing and solving various trigonometric problems.

Exploring Amplitude

Amplitude, in trigonometry, describes the height of a wave from its central axis, or midline, to the peak or trough of the wave. For the sine function, the amplitude is always a positive value.

In the standard sine formula \( y = a \sin(bx) \), the amplitude is given by \(|a|\). It determines how tall or short the wave appears on a graph.

• If \( a = 1 \), the wave reaches up to 1 and down to -1, representing its normal state.
• If \( a \) is greater than 1, the wave stretches taller.
• If \( a \) is between 0 and 1, the wave becomes shorter.
• If \( a \) is negative, it doesn't affect the amplitude but reflects the wave over the horizontal axis.

In the exercise provided, \( a \) is -2, so the amplitude is \( |-2| = 2 \). This means the wave reaches 2 units above and below the centerline.

Determining the Period

The period of a trigonometric function is the length of one complete cycle on a graph. For the sine function, it describes how far along the x-axis the function goes before it begins to repeat.

In the equation \( y = a \sin(bx) \), the period is determined by the formula \( \frac{2\pi}{b} \). This is crucial because altering \( b \) changes the wavelength of the function.

• If \( b = 1 \), the period is the standard \( 2\pi \), meaning the wave repeats every \( 2\pi \) units.
• With \( b > 1 \), the period \( \frac{2\pi}{b} \) becomes smaller, making the wave cycle more rapidly.
• If \( 0 < b < 1 \), the period increases, resulting in a slower wave cycle.

In the given example, \( b \) is \( \pi \), thus the period is \( \frac{2\pi}{\pi} = 2 \). This indicates the wave completes a full cycle every 2 units on the x-axis.