Which Of The Following Is The Graph Of $y = -3 \cdot \sin \left(\frac{1}{3} X\right)$?

by ADMIN 89 views

Introduction to Sine Waves

Sine waves are a fundamental concept in mathematics, particularly in the field of trigonometry. They are used to describe periodic phenomena, such as the motion of a pendulum or the oscillations of a spring. The general equation for a sine wave is given by y=A⋅sin⁡(Bx)y = A \cdot \sin(Bx), where AA is the amplitude and BB is the frequency. In this article, we will focus on the graph of the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) and explore its behavior.

Understanding the Graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right)

To graph the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right), we need to understand the effects of the negative coefficient and the fractional frequency on the graph. The negative coefficient −3-3 will reflect the graph across the x-axis, resulting in a downward-facing sine wave. The fractional frequency 13\frac{1}{3} will cause the graph to compress horizontally, resulting in a more rapid oscillation.

Key Features of the Graph

The graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) has several key features that are worth noting:

  • Amplitude: The amplitude of the graph is 33, which is the absolute value of the coefficient −3-3. This means that the graph will oscillate between −3-3 and 33.
  • Period: The period of the graph is 6π6\pi, which is the reciprocal of the frequency 13\frac{1}{3}. This means that the graph will complete one full cycle in an interval of 6π6\pi units.
  • Phase Shift: The graph has a phase shift of π2\frac{\pi}{2}, which is the result of the fractional frequency 13\frac{1}{3}. This means that the graph will be shifted to the right by π2\frac{\pi}{2} units.

Graphing the Function

To graph the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right), we can use a graphing calculator or software. Here is a step-by-step guide to graphing the function:

  1. Enter the function: Enter the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) into the graphing calculator or software.
  2. Set the window: Set the window to display the graph over an interval of −10π-10\pi to 10π10\pi.
  3. Graph the function: Graph the function using the graphing calculator or software.

Conclusion

In conclusion, the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is a downward-facing sine wave with an amplitude of 33 and a period of 6π6\pi. The graph has a phase shift of π2\frac{\pi}{2} and will complete one full cycle in an interval of 6π6\pi units. By understanding the behavior of sine waves and the effects of the negative coefficient and fractional frequency, we can graph the function and analyze its key features.

Key Takeaways

  • The graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is a downward-facing sine wave.
  • The amplitude of the graph is 33.
  • The period of the graph is 6π6\pi.
  • The graph has a phase shift of π2\frac{\pi}{2}.

Further Reading

For further reading on graphing trigonometric functions, we recommend the following resources:

  • Graphing Calculators: Graphing calculators are a powerful tool for graphing trigonometric functions. They can be used to graph functions, analyze their key features, and explore their behavior.
  • Math Software: Math software, such as Mathematica or Maple, can be used to graph trigonometric functions and analyze their key features.
  • Online Resources: There are many online resources available for graphing trigonometric functions, including interactive graphs and tutorials.

References

  • Graphing Trigonometric Functions: This article provides an introduction to graphing trigonometric functions, including sine waves.
  • Sine Waves: This article provides an introduction to sine waves, including their key features and behavior.
  • Graphing Calculators: This article provides an introduction to graphing calculators, including their features and uses.

Introduction

Graphing trigonometric functions is an essential skill in mathematics, particularly in the field of trigonometry. In this article, we will answer some of the most frequently asked questions about graphing trigonometric functions, including sine waves.

Q: What is the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right)?

A: The graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is a downward-facing sine wave with an amplitude of 33 and a period of 6π6\pi. The graph has a phase shift of π2\frac{\pi}{2} and will complete one full cycle in an interval of 6π6\pi units.

Q: How do I graph the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right)?

A: To graph the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right), you can use a graphing calculator or software. Here are the steps to follow:

  1. Enter the function: Enter the function y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) into the graphing calculator or software.
  2. Set the window: Set the window to display the graph over an interval of −10π-10\pi to 10π10\pi.
  3. Graph the function: Graph the function using the graphing calculator or software.

Q: What is the amplitude of the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right)?

A: The amplitude of the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is 33, which is the absolute value of the coefficient −3-3.

Q: What is the period of the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right)?

A: The period of the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is 6π6\pi, which is the reciprocal of the frequency 13\frac{1}{3}.

Q: What is the phase shift of the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right)?

A: The phase shift of the graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is π2\frac{\pi}{2}, which is the result of the fractional frequency 13\frac{1}{3}.

Q: How do I determine the amplitude, period, and phase shift of a trigonometric function?

A: To determine the amplitude, period, and phase shift of a trigonometric function, you need to analyze the equation of the function. The amplitude is the absolute value of the coefficient, the period is the reciprocal of the frequency, and the phase shift is the result of the fractional frequency.

Q: What are some common mistakes to avoid when graphing trigonometric functions?

A: Some common mistakes to avoid when graphing trigonometric functions include:

  • Incorrect amplitude: Make sure to take the absolute value of the coefficient to determine the amplitude.
  • Incorrect period: Make sure to take the reciprocal of the frequency to determine the period.
  • Incorrect phase shift: Make sure to take the result of the fractional frequency to determine the phase shift.

Conclusion

In conclusion, graphing trigonometric functions is an essential skill in mathematics, particularly in the field of trigonometry. By understanding the behavior of sine waves and the effects of the negative coefficient and fractional frequency, we can graph the function and analyze its key features. We hope that this article has answered some of the most frequently asked questions about graphing trigonometric functions.

Key Takeaways

  • The graph of y=−3⋅sin⁡(13x)y = -3 \cdot \sin \left(\frac{1}{3} x\right) is a downward-facing sine wave.
  • The amplitude of the graph is 33.
  • The period of the graph is 6π6\pi.
  • The graph has a phase shift of π2\frac{\pi}{2}.

Further Reading

For further reading on graphing trigonometric functions, we recommend the following resources:

  • Graphing Calculators: Graphing calculators are a powerful tool for graphing trigonometric functions. They can be used to graph functions, analyze their key features, and explore their behavior.
  • Math Software: Math software, such as Mathematica or Maple, can be used to graph trigonometric functions and analyze their key features.
  • Online Resources: There are many online resources available for graphing trigonometric functions, including interactive graphs and tutorials.

References

  • Graphing Trigonometric Functions: This article provides an introduction to graphing trigonometric functions, including sine waves.
  • Sine Waves: This article provides an introduction to sine waves, including their key features and behavior.
  • Graphing Calculators: This article provides an introduction to graphing calculators, including their features and uses.