Problem 1 - Precalculus: Functions and Graphs

Question

Find the amplitude and the period and sketch the graph of the equation: A. \(y=4 \sin x\) B. \(y=\sin 4 x\) C. \(y=\frac{1}{4} \sin x\) D. \(y=\sin \frac{1}{4} x\) E. \(y=2 \sin \frac{1}{4} x\) F. \(y=\frac{1}{2} \sin 4 x\) G. \(y=-4 \sin x\) H. \(y=\sin (-4 x)\)

Step-by-Step Solution

Answer

A: Amplitude 4, Period 2π; B: Amplitude 1, Period π/2; C: Amplitude 1/4, Period 2π; D: Amplitude 1, Period 8π; E: Amplitude 2, Period 8π; F: Amplitude 1/2, Period π/2; G: Amplitude 4, Period 2π; H: Amplitude 1, Period π/2.

Step 1: Understand the Basic Sine Function

The basic sine function is given by \( y = \sin x \). Its graph oscillates between -1 and 1, with an amplitude of 1 and a period of \( 2\pi \). All transformations will affect the amplitude and period based on multiplicative factors.

Step 2A: Find Amplitude and Period for A

For the equation \( y = 4 \sin x \), the amplitude is 4 as it is the coefficient of the sine function. The period is unchanged by this coefficient, so it remains \( 2\pi \).

Step 2B: Find Amplitude and Period for B

The equation \( y = \sin 4x \) modifies the period. Here, the period is \( \frac{2\pi}{b} = \frac{2\pi}{4} = \frac{\pi}{2} \), where \( b = 4 \). The amplitude remains 1 because there is no coefficient affecting the sine function.

Step 2C: Find Amplitude and Period for C

For the equation \( y = \frac{1}{4} \sin x \), the amplitude is \( \frac{1}{4} \). The period remains \( 2\pi \) since the coefficient of \( x \) is 1.

Step 2D: Find Amplitude and Period for D

In \( y = \sin \frac{1}{4}x \), the period becomes \( \frac{2\pi}{b} = \frac{2\pi}{\frac{1}{4}} = 8\pi \). The amplitude remains 1.

Step 2E: Find Amplitude and Period for E

The equation \( y = 2 \sin \frac{1}{4}x \) has an amplitude of 2 and a period of \( 8\pi \), since \( b = \frac{1}{4} \).

Step 2F: Find Amplitude and Period for F

For \( y = \frac{1}{2} \sin 4x \), the amplitude is \( \frac{1}{2} \) and the period is \( \frac{\pi}{2} \), as discussed in Step 2B.

Step 2G: Find Amplitude and Period for G

The equation \( y = -4 \sin x \) flips the sine wave across the x-axis. The amplitude is 4 and the period is still \( 2\pi \).

Step 2H: Find Amplitude and Period for H

In \( y = \sin (-4x) \), the transformation flips the graph around the y-axis, but this doesn't change the amplitude or period. Thus, the amplitude is 1 and the period is \( \frac{\pi}{2} \) as in Step 2B.

Step 3: Sketch Each Graph

For each function, sketch sine waves oscillating between positive and negative amplitudes. Consult the amplitude and period calculated in the previous steps to set the vertical stretch/compression and horizontal stretch/compression on standard graph axes.

Key Concepts

Amplitude

The concept of amplitude in a sine function refers to the height from the center line of the graph to its peak. In simpler terms, it indicates how high or low the curve stretches from the midpoint of the wave.
For a standard sine function represented as \( y = a \sin x \), the amplitude is represented by \( |a| \), which is the absolute value of the coefficient \( a \) before the sine term.

• Amplitudes can increase or decrease how wide the oscillations appear on a graph.
• They have no impact on the length of the wave along the horizontal x-axis.

For example, in the equation \( y = 4 \sin x \), the amplitude is 4, meaning the sine wave stretches four units above and below the center axis. Notice how even a negative amplitude, such as in \( y = -4 \sin x \), retains its absolute value for calculation purposes, making the amplitude 4. The negative sign indicates a vertical reflection.

Period of Sine Function

The period of a sine function is the distance required for the function to complete one full cycle along the x-axis. In a basic sine function \( y = \sin x \), the period is \( 2\pi \). However, introducing a coefficient to \( x \) changes this.
For a function of the form \( y = \sin(bx) \), the period can be calculated using \( \text{Period} = \frac{2\pi}{b} \). This formula shows how coefficients alter the speed and frequency of the sine wave's cycles.

• Increasing \( b \) causes more waves in the same interval, thus shortening the period.
• Decreasing \( b \) causes fewer waves over the same interval, lengthening the period.

For example, with \( y = \sin 4x \), the period is \( \frac{2\pi}{4} = \frac{\pi}{2} \), indicating it completes four cycles in the space \( 2\pi \) would normally host just one.

Trigonometric Transformations

Trigonometric transformations involve alterations in the sine function's amplitude, period, phase shift, and vertical shift. These transformations are expressed in the equation \( y = a \sin(bx - c) + d \).

• The coefficient \( a \) changes the amplitude and direction (vertical stretch or compression).
• The coefficient \( b \) modifies the period (horizontal stretch or compression).
• The constant \( c \) generates horizontal shifts (often called phase shifts).
• The constant \( d \) leads to vertical shifts, setting the new midpoint of the wave.

Understanding how each transformation affects the sine function helps in adjusting and predicting the graph's behavior. For instance, in \( y = \frac{1}{2} \sin(4x) \), the graph compresses vertically with an amplitude of \( \frac{1}{2} \), and horizontally with a period of \( \frac{\pi}{2} \).

Graphing Sine Functions

When graphing sine functions, it's essential to identify the amplitude, period, phase shift, and vertical shift. Begin by plotting one cycle of the wave using these key transformations:

• Identify the amplitude to determine the wave's vertical extent.
• Calculate the period to know how far along the x-axis a full wave will stretch.
• Check if any transformations like phase shifts or vertical shifts are present.

Start by marking the maximum and minimum heights using the amplitude. Then, space out the x-values according to the period. In the transformed function \( y = 2 \sin \frac{1}{4}x \), for example, the amplitude shows the vertical reach up to 2, while the extended period of \( 8\pi \) means a broader stretch across the x-axis.
Remember, sketching one full cycle accurately is the foundation. From there, extend your graph and repeat the cycle to complete the desired length on the graph.