Function \[$ M \$\] Is The Result Of A Transformation On The Parent Tangent Function.Which Equation Could Be Used To Represent Function \[$ M \$\]?A. \[$ G(x)=\tan (x+\pi) \$\]B. \[$ G(x)=\tan \left(x-\frac{\pi}{2}\right)

by ADMIN 222 views

The tangent function is a fundamental trigonometric function that has numerous applications in mathematics, physics, and engineering. In this article, we will explore the transformations of the tangent function and determine which equation could be used to represent a given function.

Understanding the Parent Tangent Function

The parent tangent function is defined as:

f(x)=tan⁡x f(x) = \tan x

This function has a period of π\pi and a vertical asymptote at x=π2+kπx = \frac{\pi}{2} + k\pi, where kk is an integer.

Transformations of the Tangent Function

The tangent function can be transformed in various ways, including:

  • Horizontal Shifts: The tangent function can be shifted horizontally by adding or subtracting a constant to the input variable xx.
  • Vertical Shifts: The tangent function can be shifted vertically by adding or subtracting a constant to the output value.
  • Reflections: The tangent function can be reflected about the x-axis or y-axis by multiplying the input variable xx or output value by −1-1.
  • Stretches and Compressions: The tangent function can be stretched or compressed horizontally by multiplying the input variable xx by a constant.

Determining the Equation of the Transformed Function

Given the function g(x)=tan⁡(x+π)g(x) = \tan (x + \pi), we can determine the equation of the transformed function by analyzing the transformation.

  • Horizontal Shift: The function g(x)=tan⁡(x+π)g(x) = \tan (x + \pi) represents a horizontal shift of the parent tangent function by π\pi units to the left.
  • Period: The period of the function g(x)=tan⁡(x+π)g(x) = \tan (x + \pi) is still π\pi, but the vertical asymptote is now at x=−π2+kπx = -\frac{\pi}{2} + k\pi.

Comparing the Options

We are given two options to represent the function g(x)g(x):

A. g(x)=tan⁡(x+π)g(x) = \tan (x + \pi) B. g(x)=tan⁡(x−π2)g(x) = \tan \left(x - \frac{\pi}{2}\right)

Option A

The function g(x)=tan⁡(x+π)g(x) = \tan (x + \pi) represents a horizontal shift of the parent tangent function by π\pi units to the left. This is consistent with the given function.

Option B

The function g(x)=tan⁡(x−π2)g(x) = \tan \left(x - \frac{\pi}{2}\right) represents a horizontal shift of the parent tangent function by π2\frac{\pi}{2} units to the right. This is not consistent with the given function.

Conclusion

Based on the analysis, the equation that could be used to represent the function g(x)g(x) is:

g(x)=tan⁡(x+π) g(x) = \tan (x + \pi)

This equation represents a horizontal shift of the parent tangent function by π\pi units to the left, which is consistent with the given function.

Final Answer

The final answer is:

g(x) = \tan (x + \pi) $<br/> **Q&A: Transformations of the Tangent Function** ============================================= In the previous article, we explored the transformations of the tangent function and determined which equation could be used to represent a given function. In this article, we will answer some frequently asked questions about transformations of the tangent function. **Q: What is the parent tangent function?** ----------------------------------------- A: The parent tangent function is defined as: $ f(x) = \tan x

This function has a period of π\pi and a vertical asymptote at x=π2+kπx = \frac{\pi}{2} + k\pi, where kk is an integer.

Q: What are the different types of transformations of the tangent function?

A: The tangent function can be transformed in various ways, including:

  • Horizontal Shifts: The tangent function can be shifted horizontally by adding or subtracting a constant to the input variable xx.
  • Vertical Shifts: The tangent function can be shifted vertically by adding or subtracting a constant to the output value.
  • Reflections: The tangent function can be reflected about the x-axis or y-axis by multiplying the input variable xx or output value by −1-1.
  • Stretches and Compressions: The tangent function can be stretched or compressed horizontally by multiplying the input variable xx by a constant.

Q: How do I determine the equation of the transformed function?

A: To determine the equation of the transformed function, you need to analyze the transformation. For example, if the function is shifted horizontally by π\pi units to the left, the equation will be:

g(x)=tan⁡(x+π) g(x) = \tan (x + \pi)

Q: What is the period of the transformed function?

A: The period of the transformed function is still π\pi, but the vertical asymptote may be shifted.

Q: How do I compare the options to determine the correct equation?

A: To compare the options, you need to analyze the transformation and determine which equation is consistent with the given function.

Q: What is the final answer?

A: The final answer is:

g(x)=tan⁡(x+π) g(x) = \tan (x + \pi)

This equation represents a horizontal shift of the parent tangent function by π\pi units to the left, which is consistent with the given function.

Frequently Asked Questions

  • Q: What is the difference between a horizontal shift and a vertical shift? A: A horizontal shift involves adding or subtracting a constant to the input variable xx, while a vertical shift involves adding or subtracting a constant to the output value.
  • Q: How do I determine the equation of the transformed function if it is reflected about the x-axis or y-axis? A: If the function is reflected about the x-axis, the equation will be:

g(x)=−tan⁡x g(x) = -\tan x

If the function is reflected about the y-axis, the equation will be:

g(x)=tan⁡(−x) g(x) = \tan (-x)

Conclusion

In this article, we answered some frequently asked questions about transformations of the tangent function. We hope this article has been helpful in understanding the transformations of the tangent function and determining the equation of the transformed function.

Final Answer

The final answer is:

g(x)=tan⁡(x+π) g(x) = \tan (x + \pi)