Problem 24 - Algebra and Trigonometry

Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function. $$y=\frac{1}{2} \sin (x+\pi)$$

Step-by-Step Solution

Answer

The amplitude of the function \( y=\frac{1}{2} \sin (x+\pi) \) is \(\frac{1}{2}\), the period is \(2\pi\), and the phase shift is \(-\pi\). Please consider all these factors when sketching the one-period graph of the function on your graph paper.

Step 1: Determine the Amplitude

The amplitude is the absolute value of the coefficient of the sin function. In this case, it is \( |\frac{1}{2}| = \frac{1}{2} \).

Step 2: Determine the Period

The period of a sine function in the form \( y=A \sin(Bx+C) \) is \(\frac{2\pi}{|B|}\). Here, our B is 1 (since x coefficient is 1), so our period would be \(\frac{2\pi}{1} = 2\pi\).

Step 3: Determine the Phase Shift

The phase shift is equivalent to the value of C in the standard sine function with negative sign i.e \(-C\). Here, \(C=\pi\) so the phase shift is \(-\pi\).

Step 4: Drawing the Graph

The standard approach to drawing the graph is to mark the amplitude, period and phase shift on the graph, and then draw a sine curve accordingly. Here, the curve begins at -π (phase shift), goes up to amplitude of 0.5, down through -0.5 and repeats every 2π (period). Please plot this on your graph paper accordingly.