Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. $$y=\tan (2 x-\pi)$$

Step-by-Step Solution

Answer

The graph of \( y=\tan(2x-\pi) \) over a two-period interval spans from 0 to \( 2\pi \), with asymptotes and zeros calculated for critical points.

Step 1: Identify the Basic Properties of the Tan Function

The basic function is \( y = \tan x \), which has a period of \( \pi \) radians. When the function is transformed to \( y = \tan(2x - \pi) \), the period will be determined by the coefficient of \( x \), which is 2 in this case.

Step 2: Calculate the Period of the Transformed Function

For the function \( y = \tan(bx) \), the period \( T \) is given by \( T = \frac{\pi}{b} \). Here, \( b = 2 \), so the period \( T \) of the function \( y = \tan (2x - \pi) \) is \( \frac{\pi}{2} \).

Step 3: Find Critical Points Within One Period

Calculate where the function reaches important values (like undefined), by setting the inside of the tangent function equal to multiples of \( \frac{\pi}{2} \), so solve \( 2x - \pi = \frac{(2n+1)\pi}{2} \) for the vertical asymptotes and \( 2x - \pi = \pi n \) for zeros, where \( n \) is an integer.

Step 4: Solve for Asymptotes and Zeros

For asymptotes, solve \( 2x - \pi = \frac{(2n+1)\pi}{2} \) to find \( x = \frac{(2n+1)\pi}{4}+\frac{\pi}{2} \). For zeros, solve \( 2x - \pi = \pi n \) to find \( x = \frac{n\pi}{2}+\frac{\pi}{2} \).

Step 5: Graph for Two Periods

To graph over two periods, calculate key values for \( n = -2, -1, 0, 1, 2, 3 \). The total interval of x-values is from \( 0 \) to \( 2\pi \) to include two period lengths of \( \frac{\pi}{2} \). Place the zeros, asymptotes, and analyze increasing/decreasing behavior between these points.

Step 6: Draw the Function

Sketch the graph using the period, asymptotes, zeros, and behavior between these points. The function should show repeating behavior every \( \frac{\pi}{2} \), with vertical asymptotes and the characteristic increasing slope of the tangent function at these intervals.

Key Concepts

Graphing Periodic Functions

Understanding how to graph periodic functions is crucial in trigonometry. These functions repeat their values in regular intervals, called periods. The function in question, given as \( y = \tan(2x - \pi) \), showcases the properties of periodic behavior. To correctly graph such functions over their specified intervals:

• Identify the period, which is the length of one complete cycle of the function.
• Determine key points within one period, such as where the function reaches maximum, minimum, or undefined values.
• Replicate the behavior of these key points over the interval to ensure accuracy.

Graphically, the function appears to repeat itself every \(\frac{\pi}{2}\), illustrating the concept of periodicity effectively.
Drawing graphs of periodic functions can initially seem challenging, but breaking it down into identifying patterns and replicating them makes it manageable. As you become more familiar with the periods and critical points, graphing becomes intuitive.

Tangent Function

The tangent function, \( y = \tan x \), is distinctive among trigonometric functions due to its unique characteristics. Unlike sine and cosine, tangent is defined as the ratio of these two functions: \(\tan x = \frac{\sin x}{\cos x}\). This relationship influences its graphical appearance:

• The period of the basic tangent function is \(\pi\), shorter than sine and cosine.
• It has vertical asymptotes where \( \cos x \) equals zero.
• The function value increases without bound and decreases similarly between these asymptotes.

The transformed function, \( y = \tan(2x - \pi) \), reflects these properties, but the period is compressed, showing the function completes a cycle in \(\frac{\pi}{2}\).
The tangent graph is particularly characterized by its steep rising and falling sections, a visual cue indicating how quickly the function's value changes.

Trigonometric Asymptotes

Trigonometric asymptotes are lines that a trigonometric function approaches but never actually reaches or crosses. In the case of the tangent function, these are vertical lines where the function is undefined. For any tangent function \(y = \tan(bx - c)\):

• Asymptotes occur at intervals of \(\frac{\pi}{b}\), adjusted by any horizontal shifts.
• Within one period, asymptotes are found at points where the inside of the tan function equals \((2n+1)\frac{\pi}{2}\). This characterizes the repeating undefined values typical of tangent graphs.

For \(y = \tan(2x - \pi)\), calculating \(2x - \pi = \frac{(2n+1)\pi}{2}\) allows us to find actual x-values for these asymptotes.
Recognizing and accurately plotting these asymptotes helps students understand why graphs of the tangent function look the way they do, adding depth to their comprehension of trigonometric behavior.

Periodicity in Trigonometry

Periodicity is a cornerstone of understanding trigonometric functions, defining their repetitive behavior over fixed intervals. With the tangent function, periodicity manifests clearly:

• The fundamental period is \(\pi\) for \(y = \tan x\), but transformations like stretching or compressing the function's argument change this.
• In \(y = \tan(2x - \pi)\), this transformed period is \(\frac{\pi}{2}\).
• This periodic nature allows us to predict and replicate function behavior beyond just one cycle.

Recognizing the periodic nature of these functions significantly aids problem-solving, as patterns and cycles inherently simplify complex calculations. Understanding periodicity not only helps in graphing but also in determining function values at various points quickly by identifying correspondences between cycles.