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

Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo

Blitzer 3rd Edition

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

Problema 3

Capitolo 2, Problema 3

Graph one period of each function. y = 2 tan x/2

Guida verificata passo dopo passo

Identify the given function: \(y = 2 \tan \left( \frac{x}{2} \right)\).

Recall the basic properties of the tangent function \(\tan x\): it has vertical asymptotes where its argument equals \(\frac{\pi}{2} + k\pi\), for any integer \(k\), and its period is \(\pi\).

Determine the period of the transformed function \(\tan \left( \frac{x}{2} \right)\) using the formula for the period of \(\tan(bx)\), which is \(\frac{\pi}{|b|}\). Here, \(b = \frac{1}{2}\), so the period is \(\pi \div \frac{1}{2} = 2\pi\).

Find the vertical asymptotes by solving \(\frac{x}{2} = \frac{\pi}{2} + k\pi\), which gives \(x = \pi + 2k\pi\). For one period, consider \(k=0\) to find asymptotes at \(x = \pi\) and \(x = -\pi\).

Plot the graph between the vertical asymptotes \(x = -\pi\) and \(x = \pi\), noting that the function is scaled vertically by a factor of 2, so the values of \(\tan \left( \frac{x}{2} \right)\) are multiplied by 2.

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Period of the Tangent Function

The standard tangent function, y = tan(x), has a period of π, meaning it repeats every π units. When the function is modified to y = tan(bx), the period changes to π/|b|. For y = 2 tan(x/2), the coefficient inside the tangent is 1/2, so the period is π divided by 1/2, which equals 2π.

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Introduction to Tangent Graph

Amplitude and Vertical Stretch

Although tangent functions do not have a maximum or minimum value, the coefficient outside the function, here 2, vertically stretches the graph. This means the output values are multiplied by 2, making the graph steeper and the values grow faster, but it does not affect the period or asymptotes.

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Stretches and Shrinks of Functions

Vertical Asymptotes of Tangent Functions

Tangent functions have vertical asymptotes where the function is undefined, occurring at points where the cosine in the denominator is zero. For y = tan(bx), asymptotes occur at x = (π/2 + kπ)/b, where k is any integer. For y = 2 tan(x/2), asymptotes are at x = π + 2kπ, guiding the graph's shape and domain restrictions.

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Asymptotes

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