Problem 16 - Algebra and Trigonometry

Question

Find the amplitude and period of the function, and sketch its graph. $$ y=-\sin 2 x $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \( \pi \). The graph is a flipped sine wave with these properties.

Step 1: Identify the Amplitude

The standard form of the sine function is \[ y = a \sin(bx) \]where \(a\) represents the amplitude. In the given function \( y = -\sin(2x) \), \(a = -1\). Amplitude is always a positive value, so the amplitude of this function is \[ |a| = 1 \].

Step 2: Determine the Period

The period of a sine function \[ y = a \sin(bx) \]is given by the formula \[ \frac{2\pi}{|b|} \].For \( y = -\sin(2x) \), \(b = 2\). Therefore, the period of this function is \[ \frac{2\pi}{2} = \pi. \]

Step 3: Sketch the Graph

Start by sketching one period of the basic sine function, \( y = \sin(x) \), with amplitude 1 and period \( \pi \). This function is flipped vertically because of the negative sign (\(-\sin(x)\)). The new function, \( y = -\sin(2x) \), has an amplitude of 1, but it compresses horizontally by a factor of 2, completing a full cycle from 0 to \( \pi \). Thus, sketch the sine wave from 0 to \( \pi \), starting from the origin, going down to -1 at \( \pi/2 \), and returning to 0 at \( \pi \).

Key Concepts

Amplitude of Sine Function

In trigonometric functions, the amplitude is a crucial concept that tells us about the vertical stretch or shrink of a sine wave. The amplitude of the sine function is determined by the coefficient in front of the sine term.
For example, in the function \( y = a \sin(bx) \), the amplitude is given by the absolute value \( |a| \). It represents how far the peaks and troughs of the wave extend from its centerline or equilibrium position.
In the specific case of \( y = -\sin(2x) \), even though there is a negative sign in front of \( \sin \), the amplitude is still positive, because amplitude is the absolute magnitude of the coefficient. Thus, here the amplitude is:

• \( |a| = |-1| = 1 \)

This means the function oscillates between 1 and -1. It showcases how the wave's peak is always 1 unit above and below the centerline, regardless of whether the function is inverted (as indicated by the negative sign).
Recognizing amplitude is essential for understanding the wave's strength and vertical extent.

Period of Sine Function

The period of a trigonometric function refers to the horizontal distance required for the function to complete one full cycle.
It is calculated using the formula \( \frac{2\pi}{|b|} \) for functions of the type \( y = a \sin(bx) \). This is because, in a standard sine wave, \( 2\pi \) radians is the full angle measurement of one complete cycle.
In the function \( y = -\sin(2x) \), the value of \( b \) is 2. Hence, the period becomes:

• \( \frac{2\pi}{2} = \pi \)

This indicates that the sine function completes one full cycle when \( x \) increases by \( \pi \) units. It shows a compression of the wave along the x-axis compared to the standard sine wave with a period of \( 2\pi \).
Understanding the period is essential for predicting where a wave pattern repeats itself and influences the frequency of these oscillations.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their amplitude, period, and any transformations like reflections.
To start, consider the basic shape of the sine function, \( y = \sin(x) \), which oscillates smoothly between 1 and -1 over a period of \( 2\pi \).
In the function \( y = -\sin(2x) \), we start by incorporating the negative sign which flips the graph vertically, inverting the usual sine wave. This flipping means that instead of reaching a peak of 1, the function starts downward, reaching -1.

• The amplitude stays at 1, affecting the peaks and troughs but not altering the flip.
• The period changes the cycle’s length to \( \pi \), compressing the wave horizontally.

The graph will start at the origin. The wave reaches the bottom at \( x = \frac{\pi}{2} \), where it hits -1, and returns to zero at \( x = \pi \).
Using these parameters to sketch, the function manifests as a single downward arch within one period. Understanding these aspects aids students in visualizing and plotting these sine variations accurately.