https://file.aiquickdraw.com/mathbot/file/1728834169893yt382fwj.jpg

The problem asks to graph the function \( y = 2 \cot(3x) \) over a one-period interval and choose the correct graph from the options.

### Step-by-step analysis:

1. **General form of cotangent function**:
   The function \( \cot(x) \) has vertical asymptotes at multiples of \( \pi \), and its period is \( \pi \). It decreases between asymptotes, crossing the x-axis at \( \frac{\pi}{2} \).

2. **Transformation for \( \cot(3x) \)**:
   - The period of the function \( \cot(kx) \) is \( \frac{\pi}{k} \).
   - For \( \cot(3x) \), the period is \( \frac{\pi}{3} \), so one complete cycle of the function occurs between 0 and \( \frac{\pi}{3} \).
   - Asymptotes occur at \( x = 0 \) and \( x = \frac{\pi}{3} \).

3. **Amplitude scaling**:
   The factor of 2 in \( 2 \cot(3x) \) only affects the vertical stretch, meaning the values of the function are multiplied by 2. This doesn't affect the position of asymptotes or zeros, only the magnitude.

4. **Zeros and asymptotes**:
   - The cotangent function crosses the x-axis midway between asymptotes. In this case, that happens at \( x = \frac{\pi}{6} \).
   - The graph will have a vertical asymptote at \( x = 0 \) and another at \( x = \frac{\pi}{3} \), with a zero at \( x = \frac{\pi}{6} \).

### Examining the options:

- The correct graph should have:
  - Vertical asymptotes at \( x = 0 \) and \( x = \frac{\pi}{3} \).
  - A zero at \( x = \frac{\pi}{6} \).
  - The graph should decrease as \( x \) increases from 0 to \( \frac{\pi}{3} \), with the amplitude stretched.

Based on these conditions, **Option A** is the correct graph because it shows the right asymptotes at \( 0 \) and \( \frac{\pi}{3} \), and the correct zero at \( \frac{\pi}{6} \), with the curve decreasing between the asymptotes.

Would you like more details on how to graph transformations, or do you have any specific questions?

### Here are 5 related questions:
1. What is the period of \( \cot(5x) \), and how would its graph look?
2. How does changing the amplitude affect the cotangent function?
3. How do we determine the vertical asymptotes of any \( \cot(kx) \) function?
4. How would the graph of \( y = -\cot(x) \) differ from that of \( \cot(x) \)?
5. How does the period of a tangent function compare to that of a cotangent function?

### Tip:
When sketching trig functions, always start by identifying the period, asymptotes, and zeros to guide the overall shape of the curve.

Graph the function y = 2 cot(3x) over a one-period interval and choose the correct graph.

Understand how to graph the function y = 2 cot(3x) over a one-period interval. This problem involves transformations of trigonometric functions, period calculations, and amplitude scaling, with a focus on identifying vertical asymptotes and zeros.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation