Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 5

Chapter 2, Problem 5

Graph two periods of the given tangent function. y = 3 tan x/4

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Identify the given function: \(y = 3 \tan \left( \frac{x}{4} \right)\).

Recall that the standard tangent function \(y = \tan x\) has a period of \(\pi\). For \(y = \tan(bx)\), the period is \(\frac{\pi}{b}\). Here, rewrite the argument as \(\frac{x}{4} = \left( \frac{1}{4} \right) x\), so \(b = \frac{1}{4}\).

Calculate the period of the function: \(\text{Period} = \frac{\pi}{b} = \frac{\pi}{\frac{1}{4}} = 4\pi\).

Since the problem asks to graph two periods, determine the interval for \(x\) that covers two periods: from \(0\) to \(8\pi\) (or any interval of length \(8\pi\)).

Plot the function \(y = 3 \tan \left( \frac{x}{4} \right)\) over the interval covering two periods, marking key points such as zeros (where the tangent argument is \(0, \pi, 2\pi, \ldots\)), vertical asymptotes (where the tangent argument is \(\frac{\pi}{2}, \frac{3\pi}{2}, \ldots\)), and the general shape scaled by the amplitude factor 3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Period of the Tangent Function

The period of the basic tangent function y = tan x is π. When the function is transformed to y = tan(bx), the period changes to π divided by the absolute value of b. Understanding how to calculate the period is essential for correctly graphing the function over the specified intervals.

Amplitude and Vertical Stretch

Although the tangent function does not have a maximum or minimum amplitude, the coefficient outside the function, such as 3 in y = 3 tan(x/4), vertically stretches the graph. This affects the steepness of the curve but does not change the period or asymptotes.

Vertical Asymptotes of Tangent

Tangent functions have vertical asymptotes where the function is undefined, occurring at x-values where the cosine is zero. For y = tan(bx), asymptotes occur at x = (π/2 + kπ)/b for all integers k. Identifying these asymptotes is crucial for accurately sketching the graph.

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