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Chapter 4: Problem 42

Sketch the graph of the function. (Include two full periods.) $$y=4 \cos x$$

Short Answer

Expert verified
The graph of the function \(y=4cosx\) is similar to the graph of \(cosx\), but oscillates between -4 and 4 instead of -1 and 1. Two full periods of the function are included in the interval from 0 to \(4\pi\).

Step by step solution

01

Understand the main characteristics of Cosine Function

The cosine function, \(cos(x)\), is a periodic function that oscillates between -1 and 1. It is important to remember that the graph of \(cos(x)\) starts at the maximum point, heads downwards to a minimum and then returns to the maximum. It has a period of \(2\pi\), meaning that \(cos(x)\) equals to \(cos(x+2\pi)\).
02

Identify the Amplitude

The amplitude of the function is the number that multiplies the cosine in the function. In this case, the amplitude is 4. Thus the function will oscillate between -4 and 4 instead of -1 and 1.
03

Sketching the basic Cosine Wave

Start by sketching the course of a basic cosine wave. Knowing that the basic cosine function wave starts at a maximum (at x=0), goes to the equilibrium point at \(\pi/2\), reaches its minimum at \(\pi\), returns to equilibrium at \((3/2)\pi\), and ends at its maximum at \(2\pi\). This is one full period.
04

Sketch the Modified Function

For the function \(y=4cosx\), apply a vertical stretch by a factor of 4 to the basic cos wave. This means that all y-values of cosine are multiplied by 4. So, the maximum becomes 4 and minimum becomes -4. Sketch one more full period from \(2\pi\) to \(4\pi\).
05

Add Labels

After sketching the two periods, add labels to the x-axis indicating the key points of each period (0, \(\pi/2\), \(\pi\), \((3/2)\pi\), \(2\pi\), \((5/2)\pi\), \(3\pi\), \((7/2)\pi\), \(4\pi\)), and to the y-axis for the maximum and minimum points (4 and -4).

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