Problem 39 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The amplitude of the function \(y=-4 \cos \frac{1}{2} x\) is 4 and the period is \(4\pi\). Its graph starts from below at -4, with maximum oscillation at 4 and it completes one period at \(4\pi\).

Step 1: Identify the Amplitude

The amplitude of a cosine function is given by the absolute value of the coefficient of \( \cos \), which here is -4. So, the amplitude is \( |-4| = 4 \).

Step 2: Identify the Period

The period of a cosine function is given by \( 2\pi/B \), where B is the coefficient that is multiplying x inside the cosine function. In this case, B is \( 1/2 \). So, the period is \( 2\pi/(1/2) = 4\pi \).

Step 3: Draw the Graph

To draw the graph of the function, start from the origin. As the coefficient of cosine is negative, the graph starts from the maximum amplitude below the x-axis (-4). The graph has to go through one complete period \(4\pi\) with amplitude of 4.