Problem 8 - Precalculus : Building Concepts and Connections

Question

Use your knowledge of horizontal translations to graph at least two cycles of the given functions. $$f(x)=\tan \left(x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\tan \left(x-\frac{\pi}{2}\right)\) is a standard tangent function, shifted right by \(\frac{\pi}{2}\) units. The graph possesses vertical asymptotes at \(x = 0\) and \(x = \pi\), and it continues to repeat this pattern for every additional cycle.

Step 1: Understand the Function

The function given is \(f(x)=\tan \left(x-\frac{\pi}{2}\right)\). This is a tangent function which has a horizontal translation (or shift) to the right by \(\frac{\pi}{2}\). The standard period of the tangent function is \(\pi\).

Step 2: Plot the Tangent Function without shift

Firstly, graph the standard tangent function, i.e., \(y=\tan(x)\). The key points to plot would be \(x = -\frac{\pi}{2}, 0, \frac{\pi}{2}\), with the function approaching \(+\infty\) as \(x\) approaches \(-\frac{\pi}{2}\), being 0 at \(x=0\), and approaching \(-\infty\) as \(x\) approaches \(\frac{\pi}{2}\). After plotting these points, remember that the function is undefined at \(x = -\frac{\pi}{2}\) and \(x = \frac{\pi}{2}\), i.e., there should be vertical asymptotes at these points.

Step 3: Apply the Horizontal Shift

The horizontal shift is applied by taking each of the x-coordinates from the standard tangent function and adding \(\frac{\pi}{2}\) to them, essentially moving the graph to the right by \(\frac{\pi}{2}\). The graph will therefore retain its vertical asymptotes, but they will be shifted to \(x = 0\) and \(x = \pi\). The function will approach \(-\infty\) as \(x\) approaches \(0\) from the right, will be 0 at \(x = \frac{\pi}{2}\), and will approach \(+\infty\) as \(x\) approaches \(\pi\). The function is cyclic and these key points repeat for each cycle with period \(\pi\).

Step 4: Graph at Least Two Cycles

With all these points in mind, you can graph two complete cycles of the function by repeating the pattern for the ranges \(x = [0, \pi]\) and \(x = [\pi, 2\pi]\).

Key Concepts

Tangent Function

The tangent function, often denoted as \( \tan(x) \), is an essential function in trigonometry. Unlike sine and cosine, which oscillate between -1 and 1, the tangent graph has no upper or lower boundaries, which means it can take on any value from negative infinity to positive infinity.

• The tangent function is periodic with a period of \( \pi \), meaning its pattern repeats every \( \pi \) units.
• It is undefined where the cosine of \( x \) is zero since the tangent function is a ratio \( \tan(x) = \frac{\sin(x)}{\cos(x)} \).
• As a result, the graph has vertical asymptotes (lines the graph gets increasingly closer to without ever touching) at every point where the cosine of \( x \) equals zero, notably at every odd multiple of \( \frac{\pi}{2} \).

Understanding these basic properties is crucial when graphing the tangent function or any transformed version of it, such as applying shifts or stretches.

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their basic shape and how they transform with different changes applied, such as horizontal translations, vertical shifts, periods, and amplitudes. For the tangent function, the changes might seem a bit different since it has no amplitude.
When graphing, start by visualizing the standard version of the function.

• For \( y = \tan(x) \), note that it passes through the origin (0,0) and moves through points where it crosses the x-axis once each period \( \pi \).
• Look for and mark the vertical asymptotes at \( x = \frac{\pi}{2} \), \( x = \frac{3\pi}{2} \), and so on, for the initial cycle.

Once the standard graph is clear, incorporate any transformations, such as a horizontal shift.

• To apply horizontal translations, adjust known points and asymptotes by the shift amount \( \frac{\pi}{2} \) to the right for the function \( f(x)=\tan \left(x-\frac{\pi}{2}\right) \). This results in asymptotes at \( x = 0 \) and \( x = \pi \).

Periodicity

Periodicity is a fundamental property of trigonometric functions, including the tangent function. Understanding this concept helps in predicting the behavior of the function over its domain. A function is periodic if there exists a positive number \( p \) such that \( f(x + p) = f(x) \) for all \( x \).
For the tangent function:

• The period is \( \pi \). This implies that its graph repeats every \( \pi \) units along the x-axis.
• Even with transformations like shifts, the period stays consistent unless explicitly altered by further modifications to the function like vertical stretching or compressing.

Understanding periodicity is key to effectively graphing and analyzing trigonometric functions because it allows one to predict graph behavior over any interval. In the case of the function \( f(x)=\tan \left(x-\frac{\pi}{2}\right) \), despite the shift, you will see the cycles repeating every \( \pi \) units, making it easier to extend the graph over different ranges, such as plotting multiple cycles as requested in the exercise.