Problem 34 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=-2 \sin (2 \pi x+\pi)\)

Step-by-Step Solution

Answer

Amplitude: 2, Period: 1, Phase shift: \(-\frac{1}{2}\) (left). Graph is a flipped sine wave.

Step 1: Identify the Amplitude

The amplitude of a sine function given by the equation \( y = a \sin(bx + c) \) is the absolute value of \( a \). Here, \( a = -2 \), so the amplitude is \( |a| = |-2| = 2 \). Therefore, the amplitude is 2.

Step 2: Determine the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \). For the equation \( y = -2 \sin(2\pi x + \pi) \), \( b = 2\pi \). Therefore, the period is \( \frac{2\pi}{2\pi} = 1 \).

Step 3: Calculate the Phase Shift

The phase shift for the sine function is given by \( \frac{-c}{b} \). In the given equation, \( c = \pi \) and \( b = 2\pi \). Thus, the phase shift is \( \frac{-\pi}{2\pi} = -\frac{1}{2} \). This is a shift of \( \frac{1}{2} \) units to the left.

Step 4: Sketch the Graph

To sketch the graph of \( y = -2 \sin(2\pi x + \pi) \), we start with the basic sine wave, apply the amplitude and reflect it because of the negative sign, adjust the period, and shift the wave left by \( \frac{1}{2} \) units. The sine wave repeats every 1 unit along the x-axis and oscillates between -2 and 2 due to the amplitude of 2, with peaks flipped due to the negative coefficient.

Key Concepts

Amplitude

In trigonometric functions, the amplitude is a measure of how far the graph of the function stretches vertically from its central axis. Specifically, for sine and cosine functions, it is half the distance between the maximum and minimum values.
For the function given by the equation \( y = a \sin(bx + c) \), the amplitude is the absolute value of \( a \). It indicates how "tall" the wave is.

• Important Note: The amplitude is always a positive value because it is a measure of distance.

In the context of this exercise, the function is \( y = -2 \sin(2\pi x + \pi) \). Here, the coefficient of the sine function, \( a \), is \(-2\). Hence, the amplitude is \( |-2| = 2 \). This tells us that the sine wave reaches 2 units above and below its central position.

Period

The period of a trigonometric function describes how long it takes for the function to complete one full cycle of its pattern. It is an essential aspect of their wave-like nature. For a function expressed by \( y = a \sin(bx + c) \), the period is found using the formula \( \frac{2\pi}{b} \).
The parameter \( b \) directly affects the frequency and period of the sine wave.

• The larger the value of \( b \), the shorter the period, which means more cycles within a given interval.
• The smaller the value of \( b \), the longer the period, meaning fewer cycles.

For this function \( y = -2 \sin(2\pi x + \pi) \), \( b = 2\pi \), leading to a period of \( \frac{2\pi}{2\pi} = 1 \). This implies that the sine wave completes one full cycle over each 1-unit interval on the x-axis.

Phase Shift

The phase shift of a trigonometric function refers to the horizontal translation of the graph along the x-axis. It alters where the wave starts its cycle. For a function \( y = a \sin(bx + c) \), the phase shift is given by the formula \( \frac{-c}{b} \). This value tells us how far, and in which direction, the graph is shifted from the usual starting point.

• A positive phase shift moves the graph to the right.
• A negative phase shift moves it to the left.

In our function \( y = -2 \sin(2\pi x + \pi) \), \( c = \pi \), so the phase shift occurs as \( \frac{-\pi}{2\pi} = -\frac{1}{2} \) units. Therefore, the graph of the sine function is shifted \( \frac{1}{2} \) units to the left.