Problem 53 - Precalculus

Question

In Exercises \(49-60\), state the amplitude, period, and phase shift (including direction) of the given function. $$y=6 \sin [-\pi(x+2)]$$

Step-by-Step Solution

Answer

Amplitude is 6, period is 2, and phase shift is 2 units to the right.

Step 1: Identify the amplitude

The amplitude of a sine function in the form \(y = A \sin(Bx + C)\) is the absolute value of \(A\). In our function \(y = 6 \sin [-\pi(x+2)]\), the amplitude \(A\) is 6. Thus, the amplitude is \(6\).

Step 2: Determine the period

The period of a sine function in the form \(y = A \sin(Bx + C)\) is calculated as \(\frac{2\pi}{|B|}\). Here, the coefficient \(B\) within the sine function is \(-\pi\), so the period is \(\frac{2\pi}{|-0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{0{{ xt{point}}}\pi|} = 2\).

Step 3: Determine the phase shift and direction

The phase shift of a sine function is calculated using the formula \(-\frac{C}{B}\). In the function provided, \(C = -2\pi\) and \(B = -\pi\). Thus, the phase shift is \(-\frac{-2\pi}{-\pi} = 2\). This positive value indicates a shift to the right by 2 units.

Key Concepts

Amplitude

In trigonometric functions, amplitude is an essential characteristic that helps us understand how much the function oscillates above and below its central axis. If you picture ocean waves, amplitude is like the height of the waves from the water's surface. For the standard form of the sine function, given by \(y = A \sin(Bx + C)\), the amplitude is represented by \(|A|\), which denotes the absolute value of \(A\). This tells you the maximum height of the curve from its central axis.In our specific example, we are given the function \(y = 6 \sin [-\pi(x+2)]\). Here, \(A = 6\). As a result, the amplitude of the function is \(6\). It’s important to note that even if \(A\) were negative, the amplitude remains positive, as it is the distance the graph stretches from its middle line.

Period

The period of a trigonometric function tells us the length of one full cycle of the curve. It is how long it takes for the function to start repeating its pattern.In the formula \(y = A \sin(Bx + C)\), the period can be found using \(\frac{2\pi}{|B|}\). This is because sine and cosine functions naturally have a period of \(2\pi\), and the coefficient \(B\) affects how tightly or loosely the curves grow.For example, in \(y = 6 \sin [-\pi(x+2)]\), the coefficient \(B\) is \(-\pi\). Therefore, the period is calculated as follows:

• \(\frac{2\pi}{|-\pi|} = 2\)

This means every 2 units along the x-axis, the sine function completes one full oscillation cycle.

Phase Shift

The phase shift of a trigonometric function depicts how the function is horizontally shifted from its regular position. It determines the starting point of the function relative to the y-axis.In the general form \(y = A \sin(Bx + C)\), the phase shift is given by the formula \(-\frac{C}{B}\). This computation provides insight into whether the curve is moved left or right.Considering our function \(y = 6 \sin [-\pi(x+2)]\), we identify \(C = -2\pi\) and \(B = -\pi\). The phase shift calculation is as follows:

• \(-\frac{-2\pi}{-\pi} = 2\)

Here, a positive result means the entire function moves to the right by 2 units along the x-axis. This phase shift allows us to clearly identify where the sinusoidal wave begins within its oscillatory cycle.