Problem 32 - Algebra and Trigonometry

Question

Determine the amplitude of each function. Then graph the function and \(y=\cos x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\). $$y=3 \cos x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=3\cos x\) is 3. When graphed alongside \(y=\cos x\), the function \(y=3\cos x\) has similar wave pattern but with a magnitude 3 times larger.

Step 1: Determine the amplitude

Amplitude is a measure of the 'height' of a wave, in a trigonometric context, it tells us how far our sine or cosine wave deviates from its mean position. For a function in the form \(y=a \cos x\) or \(y=a \sin x\), the amplitude is the absolute value of 'a'. In this case, the function is \(y=3\cos x\), so its amplitude is \(|3|=3\).

Step 2: Plot y=3cosx and y=cosx

A complete wave cycle of cosine function occurs from \(0\) to \(2\pi\). In that interval, \(\cos x\) and \(3\cos x\) achieve a maximum at \(x=0\), a minimum at \(x=\pi\), and return to maximum at \(x=2\pi\). The difference is that the maximum and minimum of \(3 \cos x\) will be 3 and -3, respectively, which is 3 times that \(cos x\). Now, the graph can be plotted by marking these critical points in the given interval [0, \(2\pi\)].

Step 3: Analyze the plotted graphs

After plotting both functions, we can see that both graphs complete a full cycle from \(0\) to \(2\pi\). However, the function \(y=3\cos x\) reaches higher up and lower down than \(y=\cos x\), reflecting its larger amplitude.