Problem 26 - Precalculus : Building Concepts and Connections

Question

Graph at least two cycles of the given functions. $$f(x)=-\tan \left(x-\frac{\pi}{3}\right)-1$$

Step-by-Step Solution

Answer

To graph the function \(f(x)= -\tan(x- \frac{\pi}{3}) -1\), understand the basic shape of the tangent function, identify transformations incorporated in the function, and apply them to the basic shape by flipping it over the x-axis, shifting it to the right by \(\frac{\pi}{3}\), and shifting it down by 1. Draw at least two of these transformed cycles on the graph.

Step 1: Identify the basic shape of the tangent function

The basic shape of the tangent function, \(\tan(x)\), is a series of vertical asymptotes and curves that cross at the origin and repeat every \(\pi\) units. Since the curve never touches the vertical asymptotes, these bounds are called the period of the function.

Step 2: Identify the transformations

The given function \(f(x) = -\tan(x - \frac{\pi}{3}) - 1\) includes three transformations: it's negated, it's shifted to the right by \(\frac{\pi}{3}\), and it's shifted down by 1. The negation will flip the function over the x-axis, the horizontal translation will shift the function to the right by \(\frac{\pi}{3}\), and the vertical translation will shift it down by 1.

Step 3: Apply the transformations to the basic shape

First, flip the basic shape of the tangent function over the x-axis. Then, shift every point to the right by \(\frac{\pi}{3}\) units. Finally, shift every point down by 1 unit. Make sure to also change the locations of the vertical asymptotes accordingly.

Step 4: Graph the function for at least two cycles

The resulting graph will have the same basic shape, but it will be flipped, shifted to the right by \(\frac{\pi}{3}\), and shifted down by 1. Also, because the tangent function repeats every \(\pi\) units, ensure to illustrate at least two cycles on the graph.

Key Concepts

Tangent Function

The tangent function, represented as \( \tan(x) \), is one of the fundamental trigonometric functions. Its graph is distinctive because it features repeating vertical asymptotes and a series of curves. Unlike sine and cosine, tangent does not have maximums or minimums, since the graph stretches infinitely in both the positive and negative direction.
The basic pattern of the tangent function repeats every \( \pi \) radians, which means its period is \( \pi \). At each half-period, there is a vertical asymptote where the function is undefined, because the tangent approaches infinity. In between these asymptotes, the curve of \( \tan(x) \) passes through the origin (0,0), moving upwards from the bottom left to the top right.
Understanding the basic shape of the tangent function is important when dealing with transformations, which will alter the way this typical graph behaves on the coordinate plane.

Function Transformations

Function transformations are operations that change the position or shape of a function's graph. In this particular exercise, we deal with three types of transformations on the function \( f(x) = -\tan(x - \frac{\pi}{3}) - 1 \): negation, horizontal shift, and vertical shift.

• Negation: Multiplying the function by -1 reflects it across the x-axis. All the curves that normally go upwards will now go downwards and vice versa.
• Horizontal Shift: Subtracting a value from \( x \) within the function moves the graph to the right by that amount. Here, \( x - \frac{\pi}{3} \) shifts the tangent function \( \frac{\pi}{3} \) units to the right.
• Vertical Shift: Subtracting a value from the entire function shifts the graph downward. In this function, subtracting 1 lowers the entire graph by 1 unit on the y-axis.

Combined, these transformations alter the appearance and location of the tangent graph from its basic form, creating a unique version of the function that fits the parameters specified in the exercise.

Graphing Trigonometric Functions

Graphing trigonometric functions accurately involves understanding their basic shapes and applying the appropriate transformations. With the tangent function \( f(x) = -\tan(x - \frac{\pi}{3}) - 1 \), graphing becomes an exercise in adjusting for the transformations we've discussed.
Firstly, as with any trigonometric function graph, it's helpful to sketch the basic shape. Then incorporate each of the transformations step by step. By flipping the graph over the x-axis, shifting it \( \frac{\pi}{3} \) units right, and then lowering it by 1, we develop a detailed understanding of how these transformations appear on the graph.
Make sure the period of \( \tan(x) \), which is \( \pi \), is considered so that you can correctly plot at least two cycles. Keep in mind the location of the asymptotes after each transformation, as these are crucial in maintaining the integrity of the tangent function's unique graph.