Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=-2 \sin (2 x-\pi)+3 $$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \( \pi \), Phase shift: \( \frac{\pi}{2} \) right.

Step 1: Identify Amplitude

The amplitude of a sine function given in the form \( y = a \sin(bx - c) + d \) corresponds to the absolute value of the coefficient \( a \). In this equation, \( a = -2 \). Therefore, the amplitude is \( |a| = |-2| = 2 \).

Step 2: Determine the Period

The period of a sine function is defined as \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) within the sine function. For our equation, \( b = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Calculate the Phase Shift

The phase shift is found by solving \( bx - c = 0 \) to find \( x \). Hence, for \( 2x - \pi \), we set it equal to zero: \[ 2x - \pi = 0 \] solving for \( x \), we get \( x = \frac{\pi}{2} \). Therefore, the phase shift is to the right by \( \frac{\pi}{2} \).

Step 4: Graphing the Function

To graph the function \( y = -2\sin(2x - \pi) + 3 \), follow these transformations:- Start with \( y = \sin(x) \).- Apply the amplitude \( -2 \): Flip across the x-axis and stretch by 2.- Apply the phase shift of \( \frac{\pi}{2} \) to the right.- Adjust for the periodicity to \( \pi \).- Finally, shift the entire wave vertically by 3 units.This gives the graph's main characteristics.

Key Concepts

Amplitude

The amplitude of a trigonometric function is an important parameter that measures the height of the wave from the center line to its peak. In the context of sine and cosine functions, the amplitude affects how "tall" the graph appears.

For a standard sinusoidal function in the form \( y = a \sin(bx - c) + d \), the amplitude is determined by the absolute value of the coefficient \( a \). In simpler terms, the amplitude is \( |a| \). This value dictates the maximum vertical distance from the sinusoidal axis (a little helper:

• If \( a \) is positive, the function retains its original wave direction.
• If \( a \) is negative, the wave is inverted or flipped vertically.

In the given equation \( y = -2 \sin(2x - \pi) + 3 \), the amplitude is \( |-2| = 2 \), indicating the wave peaks and troughs 2 units above and below the horizontal axis.

Period

The period of a trigonometric function determines how long it takes for the function to complete one full cycle. For sine and cosine functions, this is found using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \).

This value tells us how "stretched" or "compressed" the wave is.

• If \( b > 1 \), the function completes cycles more quickly, resulting in a shorter period.
• If \( b < 1 \), the function stretches out, leading to a longer period.

In our case, with \( b = 2 \), the period becomes \( \frac{2\pi}{2} = \pi \). This means that the wave repeats every \( \pi \) units along the x-axis, making it twice as "fast" as the standard \( y = \sin(x) \) function.

Phase Shift

The phase shift of a sine or cosine function refers to the horizontal movement of the graph. It indicates how much the entire wave is shifted left or right along the x-axis.

To determine the phase shift, we look at the expression inside the sine (or cosine). Specifically, from \( bx - c \), we solve for \( x \) in \( bx - c = 0 \). This gives the starting point of the wave.

• Positive values mean the graph shifts to the right.
• Negative values cause a shift to the left.

In the current function \( y = -2 \sin(2x - \pi) + 3 \), setting \( 2x - \pi = 0 \) and solving, we find that \( x = \frac{\pi}{2} \). Thus, the graph shifts to the right by \( \frac{\pi}{2} \) units, adjusting how the wave starts relative to the origin.

Graphing Trigonometric Functions

Graphing trigonometric functions involves transforming a basic wave by applying changes to its amplitude, period, phase shift, and vertical displacement. Each of these factors will guide you in sketching an accurate graph.

To graph the function \( y = -2 \sin(2x - \pi) + 3 \), follow these enhancements:

• Start with the basic graph of \( y = \sin(x) \).
• Apply the amplitude modification: because \( a = -2 \), reflect the graph vertically and stretch it by a factor of 2.
• Next, incorporate the phase shift: move the entire shape to the right by \( \frac{\pi}{2} \) units.
• Adjust for the new period: compress the wave to fit within a \( \pi \) horizontal distance.
• Finally, apply the vertical shift: move the entire graph up by 3 units to accommodate the \( +3 \) offset.

By carefully combining these transformations, you can accurately depict how the function behaves across its domain, showcasing all the key features in one coherent picture.