Problem 11 - Precalculus: Functions and Graphs

Question

Find the period and sketch the graph of the equation. Show the asymptotes. $$y=\tan 2 x$$

Step-by-Step Solution

Answer

The period of \(y=\tan 2x\) is \(\frac{\pi}{2}\), with asymptotes at \(x=\frac{\pi}{4}+n\frac{\pi}{2}\).

Step 1: Understanding the Basic Period of Tangent

The basic period of the tangent function, \( an x\), is \(rac{\pi}{3}\). However, this changes with transformations. For \( an bx\), the period is given by \(rac{\pi}{b}\). In this problem, \( an 2x\) means \(b = 2\).

Step 2: Calculating the New Period

Using the formula for the period of \( an bx\), we find that the period is \(rac{\pi}{2}\) because when substituting \(b = 2\), the expression becomes \(rac{\pi}{2}\).

Step 3: Identifying Vertical Asymptotes

The vertical asymptotes of a basic \( an x\) function occur at \(x = rac{\pi}{2} + n\pi\), where \(n\) is an integer. For \( an 2x\), the asymptotes occur at \(x = rac{\pi}{4} + n\frac{\pi}{2}\).

Step 4: Drawing the Graph

To plot \( an 2x\), draw vertical asymptotes at \(x = rac{\pi}{4}, rac{3\pi}{4}, rac{5\pi}{4}, \) etc. The function will have a typical tangent shape between these points. Each period \(rac{\pi}{2}\) will complete between two asymptotes.

Key Concepts

Periodicity of Functions

In trigonometry, periodicity refers to how functions repeat their values over regular intervals. For the tangent function, this means it cycles through its values and starts over after a certain interval. The basic period of the standard tangent function, \(\tan x\), is \(\pi\). This implies that the function repeats every \(\pi\) units along the x-axis.

When you have a transformed tangent function like \(\tan bx\), the period changes. The formula for the new period becomes \(\frac{\pi}{b}\), where \(b\) is the coefficient in front of the \(x\).

• For example, in the function \(\tan 2x\), \(b\) is 2. Thus, the period is calculated as \(\frac{\pi}{2}\).
• This means the graph of \(\tan 2x\) repeats its cycle every \(\frac{\pi}{2}\) units.

Understanding how the period of a function changes with transformations is crucial for graphing because it helps to determine how frequently the function repeats its pattern.

Graphing Trigonometric Functions

Graphing trigonometric functions, like the tangent, involves several steps to ensure accuracy. First, you need to understand the shape and typical features of the function. The tangent function, \(\tan x\), is recognizable because it increases without bound, transitioning from negative to positive quickly.

To graph \(\tan 2x\), start by analyzing its period and asymptotes. Since we've determined the period to be \(\frac{\pi}{2}\), you only need to sketch one interval of this length to show a complete cycle of the function. Key points typically used for plotting are where the function crosses the x-axis and positions of the vertical asymptotes.

• Identify where the function is undefined, leading to vertical asymptotes.
• Plot points of intersection with the x-axis, typically occurring at half of the period.

By connecting these features with the typical tangent curve shape, you can create an accurate graph of the function. The function will repeat this sketch every \(\frac{\pi}{2}\) units along the x-axis.

Vertical Asymptotes

Vertical asymptotes are lines where a function becomes unbounded, meaning it heads towards infinity or negative infinity. For trigonometric functions like tangent, these asymptotes occur at specific intervals.

In the standard \(\tan x\) function, vertical asymptotes occur at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is any integer. However, when dealing with the function \(\tan 2x\), the location of these asymptotes changes due to the transformation.

• For \(\tan 2x\), the asymptotes appear at intervals \(x = \frac{\pi}{4} + n\frac{\pi}{2}\).
• These intervals clearly show where the graph cannot pass, making them critical for accurately sketching the function.

Marking these lines on your graph is essential before drawing the curve. Knowing where the tangent goes to infinity helps dictate the overall shape and flow of your graph.