91Ó°ÊÓ

Understanding the amplitude and period of a trigonometric function is crucial for graphing it correctly. Amplitude refers to the height from the horizontal axis (or midline) to the peak of the graph. In the given equation, \(y = 2 \cos 2x\), the amplitude is 2. This means the graph stretches vertically up to 2 units above and 2 units below the midline.

The period is the distance required for the function to complete one full cycle of its pattern. For the function \(y = 2 \cos 2x\), we calculate the period using the formula \(2\pi / |b|\), where \(b\) is the coefficient of \(x\). Here, \(b = 2\) so the period is \(\pi\). This indicates every full cycle occurs over \(\pi\) units on the x-axis. Knowing these values helps plot the curve accurately, ensuring it completes a full oscillation in the specified interval.

Drawing the graph of a cosine function involves steps that standardize the process for accuracy. Begin with identifying key standard values inherent to the cosine curve:\( x = 0, \pi/2, \pi, 3\pi/2, 2\pi \). These points correspond to the maxima, zero crossings, and minima of the standard \(\cos x\) graph.

Once you've identified these points, scale them to match the specific period of the function you're working with. Our function, \(y = 2 \cos 2x\), has a period of \(\pi\), meaning these points occur at \( x = 0, \pi/4, \pi/2, 3\pi/4, \pi \).

• The maximum value occurs at \(x = 0\) and the value is 2.
• Minima occur at \(x = \pi/2\), with the value -2.
• Zeros occur at \(x = \pi/4\) and \(3\pi/4\).

Using these points, plot the characteristic peaks, troughs, and zero crossings on the graph. Connect these points smoothly into the signature wave pattern of a cosine function, maintaining the calculated amplitude and period.

The cosine function, part of the family of trigonometric functions, describes a repetitive oscillating wave. The cosine wave naturally occurs in the form \(\cos x\), which features a repetitive pattern every \(2\pi\). In a modified cosine function like \(y = 2 \cos 2x\), alterations in amplitude and frequency (or period) tailor the graph's appearance to fit specific needs.

For \(y = 2 \cos 2x\), the core characteristics change as:

• The amplitude is amplified to 2, making the top peak and bottom trough more pronounced.
• The period shortens to \(\pi\), indicating the wave completes one cycle faster than the standard \(2\pi\).

The cosine's initial maximum begins at \(x=0\), as the function value at this point is at its peak (with magnitude of the amplitude). Understanding these changes is crucial in applying the cosine function in calculations related to sound waves, light cycles, and other periodic phenomena.