Graph each function over the interval [-2π, 2π]. Give the amplitude. See Example 1.
y = 2 cos x

Verified step by step guidance

1. Identify the function type: The given function is a cosine function, specifically \( y = 2 \cos x \).
2. Determine the amplitude: The amplitude of a cosine function \( y = a \cos x \) is the absolute value of \( a \). Here, \( a = 2 \), so the amplitude is \( |2| = 2 \).
3. Set the interval for graphing: The problem specifies the interval \([-2\pi, 2\pi]\). This means you will graph the function from \(-2\pi\) to \(2\pi\).
4. Plot key points: For \( y = 2 \cos x \), identify key points within one period \([0, 2\pi]\) such as \( (0, 2), (\pi/2, 0), (\pi, -2), (3\pi/2, 0), (2\pi, 2) \) and extend this pattern to cover the interval \([-2\pi, 2\pi]\).
5. Sketch the graph: Use the key points and the amplitude to sketch the cosine wave, ensuring it oscillates between \(-2\) and \(2\) over the specified interval.

Key Concepts

Amplitude

Amplitude refers to the maximum distance a wave reaches from its central axis. In the context of trigonometric functions like cosine, it indicates how tall the peaks and how deep the troughs of the graph are. For the function y = 2 cos x, the amplitude is 2, meaning the graph oscillates between 2 and -2.

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic, with a period of 2π, meaning it repeats its values every 2π units. The graph of y = cos x is a wave that oscillates between 1 and -1, and when multiplied by a coefficient, like 2 in this case, it stretches vertically.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting their values over a specified interval, which helps visualize their periodic nature. For y = 2 cos x over the interval [-2π, 2π], one would plot points at key angles (like 0, π/2, π, etc.) and connect them to show the wave pattern. Understanding the key features such as amplitude, period, and phase shift is essential for accurate graphing.