Math Problem Statement

y=2tan3x

Solution

The equation y=2tan(3x)y = 2\tan(3x) is a trigonometric function. Let's break it down:

General Form

The tangent function, y=tan(kx)y = \tan(kx), has the following key properties:

  1. Amplitude: The tangent function does not have a defined amplitude because it increases and decreases without bound.
  2. Period: The period is given by πk\frac{\pi}{k}. Here, k=3k = 3, so the period is: Period=π3\text{Period} = \frac{\pi}{3}
  3. Vertical Stretch/Compression: The factor 22 scales the tangent function vertically, making its slope twice as steep.

Key Points

The function y=2tan(3x)y = 2\tan(3x) includes these features:

  • Period: π3\frac{\pi}{3}
  • Vertical Asymptotes: The vertical asymptotes occur where 3x=π2+nπ3x = \frac{\pi}{2} + n\pi, or x=π6+nπ3x = \frac{\pi}{6} + \frac{n\pi}{3}, where nn is an integer.
  • Scaling: The graph of tan(3x)\tan(3x) is stretched vertically by a factor of 2.

Graph Overview

The graph will:

  1. Pass through (0,0)(0, 0), since tan(0)=0\tan(0) = 0.
  2. Repeat its pattern every π3\frac{\pi}{3}.
  3. Have vertical asymptotes at x=±π6,±π2,±5π6,x = \pm\frac{\pi}{6}, \pm\frac{\pi}{2}, \pm\frac{5\pi}{6}, \dots.

Would you like me to:

  1. Graph the function?
  2. Derive its derivative y=ddx[2tan(3x)]y' = \frac{d}{dx}[2\tan(3x)]?
  3. Solve for specific values of xx?
  4. Analyze domain and range?
  5. Discuss its practical applications?

Tip: For any tangent function, understanding the period and asymptotes helps you sketch and interpret the graph more effectively.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Graphing Functions
Tangent Function Properties

Formulas

Period of tangent function: Period = π / k
Vertical asymptotes: x = (π/2 + nπ) / k

Theorems

Properties of tangent function

Suitable Grade Level

Grades 10-12