Problem 6 - Algebra 2 and Trigonometry

Question

Find the amplitude of each function. \(y=3 \sin x\)

Step-by-Step Solution

Answer

The amplitude of \(y = 3 \sin x\) is 3.

Step 1: Understand the Basic Form of the Sine Function

The basic form of the sine function is generally represented as \(y = a \sin(bx + c) + d\). In this equation, \(a\) represents the amplitude, \(b\) determines the period, \(c\) is the phase shift, and \(d\) is the vertical shift.

Step 2: Identify the Amplitude

In the given function \(y = 3 \sin x\), compare it with the standard form \(y = a \sin(bx + c) + d\). Here, \(a = 3\), which directly indicates the amplitude of the function.

Step 3: Verify the Amplitude

The amplitude of a sine function is the absolute value of \(a\). It determines how far the peaks and valleys of the wave stretch above and below the horizontal midline. Therefore, the amplitude of \(y = 3 \sin x\) is \(|3| = 3\).

Key Concepts

Amplitude

Amplitude is a key characteristic of any wave, particularly in trigonometric functions like the sine function. It represents the maximum distance a wave varies from its equilibrium position, or midpoint. In other words, it tells us how high and low the wave goes from the center line.

For the sine function, the amplitude is given by the absolute value of the coefficient in front of the sine. When you see a function such as \[y = a \sin(x)\]The amplitude is simply \[|a|\]This means, if you have a sine function, say \[y = 3 \sin(x)\]You can clearly see from the coefficient of the sine, that the amplitude is \[|3| = 3\]This amplitude shows the extent of the wave's peaks and troughs from its center line. An amplitude of 3 means the wave reaches 3 units above and 3 units below its middle point.

Sine Function

The sine function is a periodic wave that oscillates smoothly and repetitively between two values. Its inherent properties make it one of the core building blocks in trigonometry and wave mechanics.

The basic formula for the sine function is \[y = a \sin(bx + c) + d\]In this form:

• \(a\) indicates the amplitude, dictating the height of the wave.
• \(b\) influences the period, determining the length of one repetition of the wave.
• \(c\) causes the phase shift, moving the wave left or right on a graph.
• \(d\) dictates the vertical shift, impacting the mean line around which the wave oscillates.

The beauty of the sine function is its predictability and uniformity. Whether examining sound waves, light waves, or even tides, the sine function provides a mathematical basis for understanding these phenomena. Each component of the sine equation plays a vital role in shaping the wave's appearance and behavior.

Wave Properties

Understanding wave properties is essential for grasping the behavior of trigonometric functions like sine and cosine. Waves are defined by several distinctive features that determine their shape and movement.

• **Amplitude:** It measures the height of the wave's crests and the depth of its troughs from the middle line. Amplitude is one of the defining factors of a wave's size and impact.
• **Frequency and Period:** These are inversely connected. Frequency measures how many waves pass a point in a unit time, while the period is the time for one complete wave cycle. In mathematical terms, the period of a sine function \[y = a \sin(bx + c) + d\]is \[\frac{2\pi}{b}\]
• **Phase Shift:** This describes how the wave moves left or right compared to its standard position. The shift is determined by the value \(c\) in the sine function.
• **Vertical Shift:** This aspect lifts or lowers the wave relative to the center line defined by the mean or median value. This modification is specified by \(d\) in the function.

Altogether, these properties help us predict and analyze how waves behave in different scenarios, whether in physics, engineering, or everyday life.