Problem 33 - Algebra and Trigonometry

Question

\(29-42\) . Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=-4 \sin 2\left(x+\frac{\pi}{2}\right) $$

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(\pi\), Phase Shift: \(-\frac{\pi}{2}\) (left).

Step 1: Identify the General Form

The general form of a sine function is \( y = a \sin(bx + c) + d \). By comparing it to \( y = -4 \sin 2\left(x+\frac{\pi}{2}\right) \), we identify:- \( a = -4 \) (amplitude factor)- \( b = 2 \) (impacts period)- \( c = \pi \) (related to phase shift)- \( d = 0 \) (vertical shift)

Step 2: Find the Amplitude

The amplitude of a sine function \( y = a \sin(bx + c) + d \) is given by the absolute value of \( a \). Therefore, the amplitude is \( |a| = |-4| = 4 \).

Step 3: Determine the Period

The period of a sine function is calculated as \( \frac{2\pi}{b} \). For \( b = 2 \), the period becomes \( \frac{2\pi}{2} = \pi \).

Step 4: Calculate the Phase Shift

The phase shift of a sine function is determined by \( -\frac{c}{b} \). Using \( c = \pi \) and \( b = 2 \), the phase shift is \( -\frac{\pi}{2} \). This indicates a leftward shift of \( \frac{\pi}{2} \).

Step 5: Graph the Function

Start by sketching a standard sine wave with the given amplitude, period, and phase shift:- Amplitude: Peaks at 4 and troughs at -4.- Period: Repeats every \( \pi \) units.- Phase Shift: Start the wave \( \frac{\pi}{2} \) to the left of where it normally begins. This reflects across the x-axis due to the negative sign on \( a \) (-4). The steps are:1. Begin drawing a downward peak at \( x = -\frac{\pi}{2} \).2. Reach the maximum at \( x = 0 \) (since it's inverted).3. Complete one cycle back to \( x = \frac{\pi}{2} \).Complete the graph by continuing this pattern across one full period.

Key Concepts

Amplitude

The amplitude of a trigonometric function, especially a sine function, concerns the height of the peaks and depth of the troughs. In simpler terms, it's about how "tall" or "deep" the wave appears on a graph.
The amplitude is found by looking at the coefficient in front of the sine function in its general form:

• General Form: \( y = a \sin(bx + c) + d \)
• Amplitude: \(|a|\)

For the function \( y = -4 \sin 2\left(x+\frac{\pi}{2}\right)\), the amplitude is \(|-4| = 4\). This means the wave has a maximum height of 4 and a minimum depth of -4, creating a total vertical spread of 8 units.

Period

The period of a sine function defines the length of one complete cycle of the wave. It's the horizontal distance required for the function to start repeating itself over the x-axis.
This distance is derived from the coefficient \( b \) in the general sine function formula. Here's how it works:

• General Form: \( y = a \sin(bx + c) + d \)
• Period Formula: \( \frac{2\pi}{b} \)

For our example, \( b = 2 \), so the period of the function \( y = -4 \sin 2\left(x+\frac{\pi}{2}\right) \) is \( \frac{2\pi}{2} = \pi \). This indicates that the wave repeats every \( \pi \) units along the x-axis.

Phase Shift

Phase shift is all about the horizontal movement of a wave along the x-axis. It describes how much the entire graph shifts to the left or right, changing the starting point of the wave.
The phase shift is calculated using the constants \( c \) and \( b \):

• Formula: \( -\frac{c}{b} \)

When applied to our function, with \( c = \pi \) and \( b = 2 \), the phase shift becomes \( -\frac{\pi}{2} \). This means the wave moves \( \frac{\pi}{2} \) units to the left. This leftward shift alters where each cycle of the sine wave will start on the graph.

Sine Function

Sine functions are fundamental trigonometric functions that model periodic phenomena. They have a characteristic "wave-like" pattern that makes them useful for describing cyclic behaviors, such as sound waves or tidal patterns.
The standard sine function is defined mathematically as:

• General Formula: \( y = a \sin(bx + c) + d \)
• Key Components: Amplitude \(a\), Frequency \(b\), Phase Shift derived from \(c\), and Vertical Shift \(d\)

In the example \( y = -4 \sin 2\left(x+\frac{\pi}{2}\right) \):

• The negative sign before \( 4 \) indicates the wave is reflected over the x-axis.
• The function repeats itself every \( \pi \) units with an amplitude of 4.
• With a phase shift of \( -\frac{\pi}{2} \), the wave's starting point is moved left compared to the typical sine wave.

Understanding sine functions is vital for grasping more complex trigonometric concepts and their real-world applications.