Problem 14 - 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. $$ y=-\sin (3 x+\pi)-1 $$

Step-by-Step Solution

Answer

Amplitude: 1; Period: \(\frac{2\pi}{3}\); Phase Shift: \(-\frac{\pi}{3}\).

Step 1: Determine the Amplitude

The amplitude of a sine function in the form \( y = a \sin(bx + c) + d \) is given by the absolute value of \( a \). For the function \( y = -\sin(3x + \pi) - 1 \), the amplitude is \( |a| = |-1| = 1 \).

Step 2: Calculate the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \) where \( b \) is the coefficient of \( x \). In this function, \( b = 3 \). Therefore, the period is \( \frac{2\pi}{3} \).

Step 3: Find the Phase Shift

The phase shift of a sine function in the form \( y = a \sin(bx + c) + d \) is calculated as \( -\frac{c}{b} \). Here, \( c = \pi \) and \( b = 3 \). Therefore, the phase shift is \( -\frac{\pi}{3} \).

Step 4: Determine the Vertical Shift

The vertical shift is determined by \( d \) in the equation \( y = a \sin(bx + c) + d \). Here, \( d = -1 \). This means the graph will shift down by 1 unit.

Step 5: Sketch the Graph

To sketch the graph, begin by noting the amplitude of 1, a period of \( \frac{2\pi}{3} \), a phase shift of \( -\frac{\pi}{3} \) (which shifts the graph to the left), and a vertical shift down by 1 unit. Plot the sine wave considering these transformations starting from the phase shift point. Remember, since \( a = -1 \), the sine wave is inverted.

Key Concepts

Amplitude

Understanding amplitude is essential when dealing with trigonometric functions like sine and cosine. Amplitude refers to the height of the wave from its central axis to the peak. In simpler terms, it determines how tall or short the wave appears.
In the given function, \( y = -\sin(3x + \pi) - 1 \), the amplitude is extracted from the coefficient of the sine function, represented by \( a \). This coefficient is multiplied by the sine value and shows the wave's displacement.
For this function, \( a = -1 \), so the amplitude is \( |a| = |-1| = 1 \).

• The sine wave will oscillate 1 unit above and below the central axis.
• It indicates the maximum or minimum vertical distance from the wave's midline.
• Since the value is positive, amplitude is always considered in its absolute form.

This means even an inverted or reversed wave will maintain its amplitude as a positive measure of how far it extends vertically.

Period

The period of a trigonometric function relates to how often the function repeats itself over a given interval. Essentially, it is the horizontal stretch or compression of the wave. It's calculated using the formula \( \frac{2\pi}{b} \), with \( b \) being the coefficient of \( x \) in the function.
So for the equation \( y = -\sin(3x + \pi) - 1 \), \( b = 3 \). Thus, the period is calculated as \( \frac{2\pi}{3} \).

• This value tells us the length required for the sine wave to complete one full cycle.
• Higher values of \( b \) lead to more cycles within the same interval, compressing the wave.
• Conversely, a smaller \( b \) would expand each cycle over a larger space.

Therefore, in this equation, the wave will repeat every \( \frac{2\pi}{3} \) units along the x-axis.

Phase Shift

Phase shift in trigonometric functions indicates the horizontal movement of the entire wave along the x-axis. It depends on the coefficient \( c \) in the formula, calculated as \( -\frac{c}{b} \).
The function \( y = -\sin(3x + \pi) - 1 \) has \( c = \pi \) and \( b = 3 \). The phase shift is therefore \( -\frac{\pi}{3} \).
This negative value shows that the graph shifts to the left.
If \( c \) were negative, the shift would be to the right. It can alter where the cycle begins.

• Phase shifts determine the starting point of each wave cycle.
• Left movement occurs with \( +c \), while right movement with \( -c \).
• An absence of \( c \) results in no phase shift, starting at the origin.

This horizontal alteration helps fit the wave in different contexts while preserving its inherent pattern.

Vertical Shift

Vertical shifts move the wave up or down along the y-axis without altering the form of the wave. This is governed by the constant \( d \) in the function \( y = a\sin(bx + c) + d \).
In this equation, \( d = -1 \), indicating a downward movement by 1 unit.
Vertical shifts reposition the midline of the sine or cosine wave.

• Positive \( d \) values shift the wave upward.
• Negative \( d \) values move it downward.
• It affects the baseline from where the amplitude is measured.

This vertical adjustment allows for better alignment with a given set of data or required graph positioning, further customizing the sine wave's fit.