Problem 25 - Precalculus

Question

In Exercises \(9-28,\) graph the functions over the indicated intervals. $$y=\tan (2 x-\pi),-2 \pi \leq x \leq 2 \pi$$

Step-by-Step Solution

Answer

Graph the function using the period \( \frac{\pi}{2} \), shift by \( \frac{\pi}{2} \), and locate zeros/asymptotes along \(-2\pi \leq x \leq 2\pi\).

Step 1: Understanding the Function

The function given is \( y = \tan(2x - \pi) \). The basic form of the tangent function is \( y = \tan(x) \), which has a period of \( \pi \). For this function, the period is adjusted by the coefficient of \( x \).

Step 2: Determine the Period

The period of \( y = \tan(kx) \) is \( \frac{\pi}{|k|} \). In this case, \( k = 2 \), so the period is \( \frac{\pi}{2} \). This means the function repeats every \( \frac{\pi}{2} \) units along the x-axis.

Step 3: Identify the Phase Shift

The tangent function is modified by a horizontal shift or phase shift due to \( -\pi \) in the argument \( 2x - \pi \). The phase shift is given by setting the argument to zero, \( 2x - \pi = 0 \), which gives \( x = \frac{\pi}{2} \). This indicates the graph shifts right by \( \frac{\pi}{2} \).

Step 4: Define the Critical Points

The key points of the tangent function include the vertical asymptotes and the points where it crosses the x-axis. For \( y = \tan(2x - \pi) \), solve \( 2x - \pi = \frac{\pi}{2} (2n + 1) \) to find asymptotes at \( x = \frac{\pi}{4} + \frac{\pi}{2}n \). The zeros occur at \( 2x - \pi = n\pi \), giving \( x = \frac{\pi}{2}n \).

Step 5: Sketch the Graph

In the interval \(-2\pi \leq x \leq 2\pi\), sketch the graph by marking the vertical asymptotes at every \( x = \frac{\pi}{4} + \frac{\pi}{2}n \), and zeros at \( x = \frac{\pi}{2}n \). At these zeros, the function increases from negative infinity to positive infinity between each asymptote, capturing the repeating nature of the tangent function.

Key Concepts

Graphing Functions

When graphing a function, the primary aim is to provide a visual representation of an equation. This helps in understanding the nature and behavior of the function across different values. For the function \(y = \tan(2x - \pi)\), graphing over the interval \(-2\pi \leq x \leq 2\pi\) involves several key steps:

• Identify the period of the function, which affects how often the graph repeats.
• Determine any phase shifts which tell us how the graph is horizontally moved along the x-axis.
• Mark critical points such as zeros and vertical asymptotes, as they highlight significant changes in the graph.

By following these steps, you can accurately sketch the curve and observe the increasing and decreasing segments of the function.

Tangent Function

The tangent function, \(y = \tan(x)\), is a periodic function, oscillating between infinity and negative infinity, without bound. It is different from the sine and cosine functions, which are bounded. Here are a few essential characteristics of the tangent function:

• Periodicity: The tangent function has a natural period of \(\pi\), repeating the same pattern at every \(\pi\) intervals along the x-axis.
• Asymptotes: Vertical asymptotes occur where the function is undefined, creating breaks in the graph. For \(\tan(x)\), these occur at \(\frac{\pi}{2} + n\pi\), where \(n\) is an integer.
• Zeros: The function crosses the x-axis at integer multiples of \(\pi\) (e.g., \(x = 0, \pi, 2\pi, \ldots\)).

These properties are critical when examining how the tangent function has been adjusted and transformed in more complex variations like \(y = \tan(2x - \pi)\).

Period and Phase Shift

The periodicity and shifts of trigonometric functions greatly affect their graph. When you modify the standard tangent function \(y = \tan(x)\), by introducing a coefficient or a shift, it changes both the period and initial position. Here’s how these modifications work:

• Period Determination: In \(y = \tan(kx)\), the period is changed from \(\pi\) to \(\frac{\pi}{|k|}\). For \(y = \tan(2x - \pi)\), \(k = 2\), thus the period is \(\frac{\pi}{2}\). This means the function completes one full cycle every \(\frac{\pi}{2}\) units.
• Phase Shift Calculation: The expression inside the function \((2x - \pi)\) dictates any horizontal shifts. Solving \(2x - \pi = 0\) gives \(x = \frac{\pi}{2}\), indicating a shift to the right by \(\frac{\pi}{2}\). This shift affects where the new cycle of the graph begins.

These factors reshape the function's graph, showing how the function builds on its standard form by adjusting cycle lengths and starting points.

Critical Points

In graphing trigonometric functions like the tangent, understanding critical points is pivotal. These include points where the graph intercepts the x-axis, as well as steep parts where the graph approaches infinity, known as vertical asymptotes.

• Vertical Asymptotes: For \(y = \tan(2x - \pi)\), they occur at \(x = \frac{\pi}{4} + \frac{\pi}{2}n\). These asymptotes are essential as they mark regions where the graph does not exist and divide the tangent function into repeating segments.
• Zeros (Roots): Solving for \(2x - \pi = n\pi\) results in zeros at \(x = \frac{\pi}{2}n\), which is where the graph crosses the x-axis.

Highlighting these points on the graph provides a clear view of how the function behaves within a given interval, helping to easily construct and interpret the complicated layout of these graphical representations.