Problem 10 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=2 \sin \frac{1}{4} x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=2 \sin \frac{1}{4} x\) is 2, and the period is \(8\pi\). One period of the function shows a sine wave shape spanning from x = 0 to x = \(8\pi\), and a height from y = -2 to y = 2.

Step 1: Identify the Amplitude

The coefficient of the sine function, |A|, is the amplitude of the function. In this function, A = 2, so the amplitude is |2| = 2.

Step 2: Identify the Period

The period (T) of the function is given by \( T = \frac{2\pi}{|B|}\). In this sine function, B = 1/4, so plugging this into the formula gives \( T = \frac{2\pi}{|1/4|} = 8\pi\).

Step 3: Sketch the graph

With the amplitude and period known, proceed to sketch the graph of the function. Start at the origin (0, 0), then plot points at key positions around the period (T = 8π) while respecting the amplitude (2). The points are (0,0), (2π,2), (4π,0), (6π,-2) and (8π,0). After connecting these points smoothly, a single period of the function will be completed.