Problem 164 - Precalculus Essentials

Question

Determine the amplitude, period, and phase shift of \(y=4 \sin (2 \pi x+2) .\) Then graph one period of the function. (Section 4.5, Example 4 )

Step-by-Step Solution

Answer

The amplitude of the function \(y=4 \sin (2 \pi x+2)\) is 4. The period of the function is 1. The phase shift of the function is 1. The graph of one period of the function will have these characteristics.

Step 1: Find the Amplitude

The amplitude of a sine function is the absolute value of the coefficient of the sine function. In the function \(y=4 \sin (2 \pi x+2)\), the amplitude, A, is \(|4|\), which is 4.

Step 2: Find the Period

The period, P, is found by dividing \(2\pi\) by the absolute value of B. In the function \(y=4 \sin (2 \pi x+2)\), notice that B is \(2\pi\). That means, \(P= 2\pi /|2\pi|\), which simplifies to \(P=1\). Therefore, the period of the function is 1.

Step 3: Find the Phase Shift

The phase shift is found by taking the C value (if any) and evaluating how it shifts the standard sine wave along the x-axis. In \(y=4 \sin (2 \pi x+2)\), C is 1. Thus, the phase shift is 1. This value tells us how much our sine function is shifted horizontally.

Step 4: Graph the Function

To plot the function, first mark the period from 0 to 1 along the x-axis. Since it is a sine function, it starts at 0, goes up to the amplitude at \(P/4\), comes back to 0 at \(P/2\), goes down to the negative of the amplitude at \(3P/4\) and returns to zero at the end of the period. With a phase shift, simply shift each of these key points one unit to the right. Finally, plot the points and sketch the curve.

Key Concepts

Amplitude

Amplitude in the context of trigonometric functions, like a sine function, is a measure of the maximum distance of the wave from its equilibrium position. It's essentially how "tall" or "short" the wave is. In our given function, \(y=4 \sin (2 \pi x+2)\), the amplitude can be identified by looking at the coefficient of the sine term, which is 4.

• Amplitude is always a positive value. Even if the coefficient is negative, you take the absolute value.
• In a graph, amplitude determines the height from the centerline (usually the x-axis) to the peak of the wave.

Understanding amplitude helps us visualize how vertical stretches or shrinks affect the shape and size of a sine wave, making it an essential part of graphing trigonometric functions.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle. It's represented by "P" and tells us the length of one repetition of the wave along the x-axis. In this function \(y=4 \sin (2 \pi x+2)\), the period can be found by looking at the number multiplying \(x\), which is \(2\pi\) in this case.

• Formula: The period \(P\) is calculated as \(\frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\).
• In our case, since \(B = 2\pi\), we calculate \(P = \frac{2\pi}{|2\pi|} = 1\).

For students, knowing the period is crucial as it aids in understanding how functions behave over a set interval and helps in correctly plotting the wave on graphs.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric graph along the x-axis. It shows us how the start of the wave is shifted from the usual position. In the function \(y=4 \sin (2 \pi x+2)\), the phase shift is derived from the expression inside the sine function. Specifically, it depends on the constant term added or subtracted from \(x\).

• The general formula for phase shift in a function \(A \sin(Bx + C)\) is \(-\frac{C}{B}\).
• For our function, with \(B = 2\pi\) and \(C = 2\), we find the phase shift using \(-\frac{2}{2\pi} = -\frac{1}{\pi}\).

Understanding the phase shift helps in accurately positioning the wave on the graph, crucial for interpreting and predicting function behavior under different transformations.