Problem 36 - Algebra and Trigonometry

Question

Determine the amplitude and period of each function. Then graph one period of the function. $$y=\cos 4 x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=cos(4x)\) is 1. The period of this function is \(\pi/2\).

Step 1: Identify the Amplitude

The amplitude of any sine or cosine function is the absolute value of the coefficient in front of the trigonometric function. In this case, the function is \(y=\cos 4x\), so the coefficient, and thus the amplitude, is 1.

Step 2: Calculate the Period

The period of the function can be found by the formula \(2\pi/|b|\), where 'b' is the coefficient of x inside the trigonometric function. Here, \(b=4\), therefore, the period of the function is \(\pi/2\) or \(1.57\) approx.

Step 3: Graph the Function

Start by marking the period and amplitude on the x- and y-axis respectively. The graph starts at the point (0,1) because cosine of 0 is 1. A full cycle of the cosine function from 0 to \(\pi/2\) consists of one peak and one trough. At x=\(\pi/4\), y=0 and At x=\(\pi/2\)