Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. State the phase shift. $$y=2 \cos (x+\pi)$$

Step-by-Step Solution

Answer

The phase shift is \(\pi\) units to the left.

Step 1: Identify the function type

The given function is a cosine function, specifically given as \( y = 2 \cos(x + \pi) \). This is a standard trigonometric function with a modification to its phase.

Step 2: Determine amplitude, period, and phase shift

The amplitude of the function is the coefficient in front of the cosine, which is 2. The standard period of a cosine function \( \cos(x) \) is \(2\pi\). The phase shift is determined by the horizontal translation component in the angle, which here is \(+\pi\). This implies the graph shifts to the left by \(\pi\) units.

Step 3: Graph the function over two periods

To graph \( y = 2 \cos(x + \pi) \) over two cycles, begin by noting that the period is \(2\pi\). Therefore, two complete periods cover \([0, 4\pi]\). The phase shift of \( \pi \) means the starting point of one cycle is at \(-\pi\) and the ending point of the second cycle is at \(3\pi\). Plot points at quarter-period increments starting from \(-\pi\), then sketch the cosine wave accordingly.

Step 4: State the phase shift

By analyzing the function \( y = 2 \cos(x + \pi) \), the phase shift is determined by the term \( x + \pi \). This indicates a shift to the left by \(\pi\) units.

Key Concepts

Amplitude

When exploring trigonometric functions, amplitude is a core concept to grasp. It represents the height of the wave from its central axis to its peak. In the function \( y = 2 \cos(x + \pi) \), the amplitude is 2. This indicates that the cosine wave reaches 2 units above and below its mean position.

• The amplitude is always the absolute value of the coefficient of the cosine or sine term. It dictates the ‘stretch’ or ‘shrink’ of the graph vertically.
• If the amplitude had been negative, such as -2, while the highest and lowest points would still be 2 units from the center, the wave would be reflected across the horizontal axis.

Understanding amplitude helps in envisioning the graph's appearance without plotting. It’s crucial for predicting the oscillation behavior of a trigonometric function.

Period

The period of a trigonometric function defines how long it takes for the wave to complete one full cycle. For a standard cosine function like \( \cos(x) \), this period is \(2\pi\). It means that every \(2\pi\) units along the x-axis, the wave pattern repeats itself.

• In the function \( y = 2 \cos(x + \pi) \), the period remains at \(2\pi\) since there is no stretching or compression horizontally.
• This property is crucial for determining where key points such as peaks, troughs, and zero crossings occur on the graph.

By knowing the period, you can accurately sketch the function over any interval by repeating the pattern every \(2\pi\). It helps predict how the wave continues in either direction along the x-axis.

Phase Shift

Phase shift describes the horizontal movement of the trigonometric wave on a graph. It's defined by the horizontal translation factor in the function\( (x + \pi) \). For our function \( y = 2 \cos(x + \pi) \), we have a phase shift because of the \( +\pi \) added to x.

• This particular translation is a shift to the left by \(\pi\) units.
• Understand phase shift as a way to control where the wave starts on the x-axis. It's like sliding the entire wave to either left or right.

The phase shift helps adjust where cycle beginnings, such as peaks or zeros, appear. It’s essential for aligning the wave with particular points or features on the graph.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented as \( y = \cos(x) \). The graph of this function is a smooth wave that oscillates between -1 and 1, repeating every \(2\pi\) units.

• Variations of the basic form, such as \( y = 2 \cos(x + \pi) \), modify these fundamental attributes.
• The coefficient before the cosine determines the amplitude, and additional terms in the angle dictate the phase shift and period changes.

Recognizing a cosine function helps predict its general behavior, such as periodicity, symmetry, and oscillation. By identifying certain parameters, like amplitude or phase shift, you can quickly anticipate how its graph will look and function.