Problem 36 - Algebra 2 and Trigonometry

Question

Sketch one cycle of each function. \(y=-\cos \frac{1}{2} x\)

Step-by-Step Solution

Answer

Graph a cosine wave starting from the lowest point, and complete one period within \([0, 2\pi]\).

Step 1: Identify the amplitude

The amplitude of a cosine function, given by the coefficient before the cosine, determines how far the graph oscillates above and below the midline. In the function \(y = -rac{1}{2} \cos x \), the amplitude is the absolute value of the coefficient before the cosine, which is \(-1\). Thus, the amplitude \(A\) is 1.

Step 2: Determine the period

The period of a cosine function is calculated using \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\). In our function, \(B=1\), leading to \(\frac{2\pi}{1} = 2\pi\), implying that the period of the function is \(2\pi\).

Step 3: Consider the vertical flip

The negative sign in front of \(\cos x\) indicates a reflection of the graph across the x-axis. Therefore, where the standard cosine function starts at the maximum point in a positive graph, this function starts at the minimum point because of the reflection.

Step 4: Plot key points

Plot critical points of the cosine curve over one period \([0, 2\pi]\): begin at \((0, -1)\), reach 0 at \((\frac{\pi}{2}, 0)\), maximum at \((\pi, 1)\), return to 0 at \((\frac{3\pi}{2}, 0)\), and complete the cycle at \((2\pi, -1)\).

Step 5: Draw the curve

Connect these key points smoothly to form the characteristic 'wave' shape, which will appear as a 'flipped' cosine wave due to the negative sign, resulting in one complete cycle of the function \(y = -\cos x\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function that plays a crucial role in various mathematical and scientific applications. It's often represented as \(\cos x\). When graphing the cosine function, we observe a distinct wave-like pattern that repeats consistently. This is due to its periodic nature.

Key characteristics of the cosine function include:

• It begins at its maximum value of 1 when the angle \(x\) is 0.
• As \(x\) increases, the value of \(\cos x\) decreases, reaching 0 at \(\pi/2\), then drops to its minimum value of -1 at \(\pi\).
• After \(\pi\), the pattern symmetrically ascends to 0 at \(3\pi/2\) and peaks at 1 again at \(2\pi\).

For the cosine function \(y = -\cos x\), there is a vertical reflection due to the negative sign in front of the cosine. Therefore, all the values of the customary cosine function are mirrored about the x-axis. Instead of starting at a maximum, it starts at a minimum value. This reflection results in the graph starting at \(-1\), reaching 0 at \(\pi/2\), peaking at 1 at \(\pi\), and so forth.

Period of Trigonometric Functions

The period of a trigonometric function tells us how long it takes for the function to complete one full cycle of its pattern. For the basic cosine function \(y = \cos x\), this cycle length, known as the period, is \(2\pi\). This means that every \(2\pi\) units along the x-axis, the cosine function repeats its shape.

If we have a modified cosine function, such as \(y = -\cos\frac{1}{2}x\), the period of the function is affected by the coefficient of \(x\). We can calculate this new period by using the formula:

• Period = \(\frac{2\pi}{B}\)
• Where \(B\) is the coefficient of \(x\).

In the example \(y = -\cos\frac{1}{2}x\), the period becomes \(\frac{2\pi}{1/2}\), simplifying to \(4\pi\). Consequently, it takes \(4\pi\) to complete one full cycle, which is double the typical period of a regular cosine function due to adjusting the frequency with \(\frac{1}{2}\). This elongates the wave, creating wider arcs as seen in its graph.

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function defines the height of its peaks and depths of its troughs from the function's midline. For a basic cosine function like \(y = \cos x\), the amplitude is 1. This implies the graph oscillates between 1 and -1.

This concept is quite straightforward: the amplitude is the absolute value of the coefficient in front of the cosine term. In a function like \(y = -\cos x\), the amplitude remains positive despite the negative sign because amplitude is always considered as a positive quantity. Thus, the amplitude is:

• Amplitude = \(\left|a\right|\) where \(a\) is the coefficient of the cosine function.

In the example of \(y = -\cos \frac{1}{2}x\), the amplitude is likewise 1. This means the wave's crests and troughs will reach 1 unit above and below its central axis, the x-axis in this instance. The negative sign simply reflects the wave over the x-axis, yet doesn't alter the amplitude. It's essential to remember, amplitude measures magnitude, not direction.