Problem 8 - Precalculus: A Concise Course

Question

Find the period and amplitude. $$ y=-3 \sin \frac{x}{3} $$

Step-by-Step Solution

Answer

The amplitude is 3 and the period is \(6\pi\).

Step 1: Identifying the Amplitude

The amplitude of a sine function is given by the absolute value of the coefficient of the sine term. In this case, the coefficient is -3, so the amplitude will be |-3| = 3.

Step 2: Finding the period

The period of a sine function is given by the formula \( \frac{2\pi}{|b|}\), where 'b' is the coefficient of the x term in the sine function. In this case, since our function is \( \sin \frac{x}{3}\), the b value is \( \frac{1}{3}\). Therefore, substituting into the formula we get \( \frac{2\pi}{|\frac{1}{3}|}= 6\pi\).

Key Concepts

Amplitude

Amplitude in trigonometric functions refers to the maximum distance the wave reaches from its central axis. It measures how "tall" or "short" the wave is.
In the sine function, typically expressed as \(y = A \sin(Bx + C) + D\), the amplitude is determined by the value \(A\).

• If \(A\) is positive, the wave "starts" going up from the axis.
• If \(A\) is negative, it "starts" going down.
• The amplitude is always a positive value, as it's calculated using the absolute value, \(|A|\).

In the given function \(-3 \sin \frac{x}{3}\), the amplitude is determined from \(-3\).
The absolute value tells us \(|-3| = 3\). This means the wave stretches 3 units above and below its central line.

Period

The period of a trigonometric function like the sine function quantifies the length of one complete wave cycle on the x-axis.
The standard sine function \( \sin(x) \) completes one cycle every \(2\pi\) radians. Other variations, like \( \sin(Bx) \,\), change this frequency.

• The formula to find the period of \( \sin(Bx) \) is \( \frac{2\pi}{|B|} \).
• The value \(B\), known as the frequency, affects how "stretched" or "compressed" the wave is.

In our specific case, the function is \( \sin \frac{x}{3} \).
Here, \(B = \frac{1}{3}\).
Plug this into our period formula: \( \frac{2\pi}{|\frac{1}{3}|} = 6\pi \).
This tells us one full cycle of the wave spans \(6\pi\) units along the x-axis.

Sine Function

The sine function is a fundamental wave-like function in trigonometry. It is often expressed as \( y = A \sin(Bx + C) + D \), describing a wave with specific characteristics.
The sine function is periodic, meaning it repeats its pattern endlessly along the x-axis.

• It starts at zero, rises to a peak (positive amplitude), returns to zero, falls to a low point (negative amplitude), and rises back to zero.
• The characteristic shape is smooth and oscillating.
• In terms of transformations, the coefficients \(A\), \(B\), \(C\), and \(D\) alter its graph as follows:
• \(A\) changes the amplitude (height).
• \(B\) changes the period (width).
• \(C\) shifts the graph horizontally (phase shift).
• \(D\) shifts it vertically.

The function \( y = -3 \sin \frac{x}{3} \) is a transformed sine function.

• Here, \(A = -3\) alters the amplitude and inverts the wave.
• The term \( \frac{1}{3}x \) changes the period, stretching it to \(6\pi\).

This example illustrates how trigonometric functions can model real-world waves, like sound or light.