Problem 36 - Precalculus : Building Concepts and Connections

Question

Graph at least two cycles of the given functions. $$h(x)=2|\cos (3 x)|-1$$

Step-by-Step Solution

Answer

Plot the coordinates (0,-1), (\(\pi / 6\),1), (\(\pi / 3\),-1), etc, up to (\(4\pi / 3\),-1). Connect these points with smooth, wavelike curves to form at least two cycles of the function \(h(x) = 2|\cos (3x)| - 1\).

Step 1: Determine the Main Function

It's essential to recognize the main function, which is the absolute value function. The function within the absolute value bars is \(2\cos(3x) - 1\), a simple cosine function that's been scaled and shifted.

Step 2: Identify the Period

The period of a cosine function is \(2\pi\). However, our function has been scaled by a factor of 3, which changes the period. The period of our function is \(2\pi / 3\). We'll need to graph at least two periods, or a range from \(0\) to \(4\pi / 3\).

Step 3: Plot the Function

To plot the function, start at \(x = 0\) and plot points at each quarter period, or \(\pi / 6\), until reaching \(x = 4\pi / 3\). The points will follow the pattern of a stretched and reflected cosine function, oscillating between -1 and 1.

Key Concepts

Absolute Value Transformations

When dealing with trigonometric functions, introducing absolute value can significantly alter their graphs. In the context of the function presented, which is \(h(x)=2|\cos(3x)|-1\), the absolute value operator influences the function's output so that it only yields non-negative values.

Normally, a cosine wave crosses the x-axis, producing both positive and negative values. However, when the cosine function is enclosed in absolute value bars, any negative output is reflected above the x-axis. This transformation results in a waveform that pulses entirely above the x-axis, or, in this particular case, above the horizontal line \(y = -1\) due to the additional subtraction.

Understanding this concept is crucial, as it impacts not just the graph's general shape but also its amplitude – the height of the peaks from the central line of oscillation. For \(h(x)\), you would expect to see peaks at a height of \(1\) above the horizontal line \(y = -1\), giving a maximum value of \(0\) due to the absolute value effect.

Cosine Function Period

In trigonometry, one important characteristic of a cosine function is its period, which is the length of one complete cycle of the wave. The standard period of a cosine function, \(\cos(x)\), is \(2\pi\). However, when the cosine function is multiplied by a factor within its argument, as seen in \(\cos(3x)\), the period gets adjusted.

To find the new period of the function, divide the standard period by the absolute value of the factor. In this case, the new period is \(\frac{2\pi}{3}\). This alteration means that the function completes one full cycle three times as fast compared to the basic cosine function. Graphing at least two cycles of \(h(x)\), as the exercise suggests, requires plotting the function over a span of \(\frac{4\pi}{3}\), which ensures that two full 'new' periods are covered.

Amplitude and Phase Shift

Amplitude is the measure of how far the peaks of the wave are from its central axis; it defines the wave's height. Normally, the amplitude is half the distance between the maximum and minimum values of the function. For the function \(h(x)\), the amplitude can be directly observed as the coefficient in front of the cosine function prior to the absolute value transformation, which is \(2\). This number tells us that the height of the peaks from the central line of oscillation is 2 units for the original cosine wave, but after the absolute value transformation, the peak only reaches 1 unit above the horizontal line \(y = -1\).

Phase shift, on the other hand, refers to the horizontal displacement of the trigonometric function along the x-axis. For \(h(x)=2|\cos(3x)|-1\), there isn't an explicit horizontal shift, as there is no addition or subtraction within the argument of the cosine function. However, a vertical shift downwards is present due to the subtraction of 1, which affects the central line of oscillation.

When graphing trigonometric functions, understanding these concepts is key to accurately representing the shift and size of the waves, ensuring a correct interpretation of the function's behavior.