Problem 33 - Precalculus Essentials

Question

Determine the amplitude of each function. Then graph the function and \(y=\cos x\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi\) $$y=-2 \cos x$$

Step-by-Step Solution

Answer

The amplitude of the function \(y=-2 \cos x\) is 2. Its graph within the given range is a cosine wave vertically stretched by factor of 2 and reflected over the x-axis. When compared to the standard cosine function \(y = \cos x\), the peaks and valleys are double in magnitude and opposite in direction.

Step 1: Determine the amplitude

The amplitude of a function is found by taking the absolute value of its coefficient. In this case, the coefficient of the cosine function in \(y = -2 \cos x\) is -2. When we take the absolute value of -2, we get 2. Therefore, the amplitude of the function \(y = -2 \cos x\) is 2.

Step 2: Graph the function \(y=-2 \cos x\)

Next, we need to graph the function \(y=-2 \cos x\). This function will yield a cosine wave which is vertically stretched by a factor of 2 (due to the amplitude of 2) and reflected over the x-axis (due to the negative sign). It will peak at y=-2 and valley at y=2. Use a graphing calculator or software and plot this function for the range \(0 \leq x \leq 2 \pi\). Note the key points: at \(x=0\), it starts at y=-2; at \(x=\pi/2\), y=0; at \(x=\pi\), y=2; at \(x=3\pi/2\), y=0; and at \(x=2\pi\), y=-2.

Step 3: Graph the function \(y=\cos x\)

Next, we will graph the standard cosine function \(y=\cos x\) within the same coordinate system for comparison. This function will yield a standard cosine wave which peaks at y=1 and valleys at y=-1. Important points to note for plotting: at \(x=0\), it starts at y=1; at \(x=\pi/2\), y=0; at \(x=\pi\), y=-1; at \(x=3\pi/2\), y=0; and at \(x=2\pi\), y=1.

Key Concepts

Amplitude

In the world of trigonometric functions, amplitude is a fundamental concept, particularly when dealing with sine and cosine waves. Amplitude refers to the height of the wave from its centerline to its peak (or trough). It gives us an idea of how "tall" or "short" a wave is.

• In mathematical terms, the amplitude of a function such as \(y = A \, \cos(x)\) is the absolute value of A.
• For the function \(y = -2 \cos x\), the coefficient in front of the cosine function is \(-2\). The amplitude here is \(|-2| = 2\).

This means the curve will stretch vertically by a factor of 2, producing peaks at a distance of 2 above and 2 below the midline.
Understanding amplitude is crucial because it affects the intensity of the wave but does not alter its period or frequency.

Graphing Trigonometric Functions

When we graph trigonometric functions like \(y = \cos x\) or \(y = -2 \cos x\), we visualize these mathematical concepts and their characteristics. The graph of a typical cosine function features a smooth, wave-like pattern oscillating between a maximum and minimum value.

• The standard cosine curve \(y = \cos x\) oscillates with peaks at \(+1\) and troughs at \(-1\).
• For \(y = -2 \cos x\), the graph reflects over the x-axis due to the negative sign, and the peaks occur at \(-2\) while troughs occur at \(+2\).

Key points on the cosine graph such as \(x = 0, \pi/2, \pi, 3\pi/2, \, \text{and} \, 2\pi\) are instrumental in outlining its shape. These act as guides for plotting, indicating where the curve touches the x-axis or reaches extreme values. Whether you're using graphing calculators or plotting by hand, these points help in drawing the function accurately.

Trigonometric Transformations

Trigonometric transformations involve changes to the basic trigonometric graphs to produce variations of the standard waves. Such transformations might involve reflections, stretches, compressions, or translations along the axes. By understanding these transformations, we can predict how a trig function will look and behave differently from its prototype.

• A reflection across the x-axis happens when there's a negative sign in front of the function, like \(y = -2 \cos x\).
• Vertical stretching or compressing occurs when the amplitude changes from the standard value. Thus, \(y = -2 \cos x\) is a vertically stretched version of \(y = \cos x\).

These transformations do not change the period or the base frequency but do modify the visual and spatial trajectory of the wave on the graph. Recognizing these transformations is key to analyzing and sketching such functions effectively.