In Exercises 31–34, determine the amplitude of each function. Then graph the function and y = cos x in the same rectangular coordinate system for 0 ≤ x ≤ 2π. y = 2 cos x

Verified step by step guidance

1. Identify the general form of the cosine function, which is \(y = A \cos x\), where \(A\) represents the amplitude.
2. Recall that the amplitude of a cosine function is the absolute value of the coefficient in front of \(\cos x\), so amplitude \(= |A|\).
3. For the given function \(y = 2 \cos x\), determine the amplitude by taking the absolute value of 2, which is \(|2|\).
4. To graph the function \(y = 2 \cos x\) along with \(y = \cos x\) on the same coordinate system for \(0 \leq x \leq 2\pi\), plot points for both functions at key values of \(x\) such as \(0\), \(\frac{\pi}{2}\), \(\pi\), \(\frac{3\pi}{2}\), and \(2\pi\).
5. Note that the graph of \(y = 2 \cos x\) will have peaks at \(2\) and troughs at \(-2\), while \(y = \cos x\) has peaks at \(1\) and troughs at \(-1\), reflecting the difference in amplitude.

Key Concepts

Amplitude of a Trigonometric Function

Amplitude is the maximum absolute value of a trigonometric function from its midline. For functions like y = a cos x, the amplitude is |a|, representing the peak height of the wave above or below the horizontal axis.

Graphing the Cosine Function

The cosine function y = cos x is periodic with period 2π, oscillating between -1 and 1. Understanding its shape, key points (0, 1), (π, -1), and (2π, 1), and symmetry helps in accurately plotting it on a coordinate system.

Comparing Multiple Functions on the Same Graph

Plotting y = 2 cos x alongside y = cos x requires understanding how amplitude affects the graph's vertical stretch. Comparing both on the same axes highlights differences in amplitude while sharing the same period and phase.