Select The Correct Answer.Which Sequence Of Transformations Was Applied To The Parent Tangent Function To Create The Function $m(x)=2 \tan (3x+4$\]?A. Horizontal Compression By A Factor Of $\frac{1}{3}$, Phase Shift Left

Introduction

The tangent function is a fundamental concept in trigonometry, and its transformations play a crucial role in various mathematical applications. In this article, we will explore the sequence of transformations applied to the parent tangent function to create the function $m(x)=2 \tan (3x+4)$. We will analyze each transformation step by step to understand how the parent function is modified to obtain the given function.

Parent Tangent Function

The parent tangent function is given by $f(x) = \tan x$. This function has a period of $\pi$ and a vertical asymptote at $x = \frac{\pi}{2} + k\pi$, where $k$ is an integer.

Transformation 1: Horizontal Compression

The first transformation applied to the parent tangent function is a horizontal compression by a factor of $\frac{1}{3}$. This means that the function is stretched horizontally by a factor of $3$. To achieve this, we replace $x$ with $\frac{x}{3}$ in the parent function.

f(x) = \tan x \rightarrow f(x) = \tan \left(\frac{x}{3}\right)

Transformation 2: Phase Shift Left

The second transformation applied to the parent tangent function is a phase shift left by $\frac{4}{3}$. This means that the function is shifted to the left by $\frac{4}{3}$ units. To achieve this, we replace $x$ with $x - \frac{4}{3}$ in the function obtained after the horizontal compression.

f(x) = \tan \left(\frac{x}{3}\right) \rightarrow f(x) = \tan \left(\frac{x - \frac{4}{3}}{3}\right)

Transformation 3: Vertical Stretch

The third transformation applied to the parent tangent function is a vertical stretch by a factor of $2$. This means that the function is stretched vertically by a factor of $2$. To achieve this, we multiply the function obtained after the phase shift left by $2$.

f(x) = \tan \left(\frac{x - \frac{4}{3}}{3}\right) \rightarrow f(x) = 2 \tan \left(\frac{x - \frac{4}{3}}{3}\right)

Final Function

The final function obtained after applying all the transformations is $m(x) = 2 \tan (3x + 4)$. This function has a period of $\frac{\pi}{3}$ and a vertical asymptote at $x = \frac{2\pi}{3} + k\frac{\pi}{3}$, where $k$ is an integer.

Conclusion

In this article, we have analyzed the sequence of transformations applied to the parent tangent function to create the function $m(x) = 2 \tan (3x + 4)$. We have seen how each transformation modifies the parent function to obtain the given function. Understanding these transformations is essential in various mathematical applications, including calculus and differential equations.

Key Takeaways

• The parent tangent function is given by $f(x) = \tan x$.
• The first transformation applied to the parent tangent function is a horizontal compression by a factor of $\frac{1}{3}$.
• The second transformation applied to the parent tangent function is a phase shift left by $\frac{4}{3}$.
• The third transformation applied to the parent tangent function is a vertical stretch by a factor of $2$.
• The final function obtained after applying all the transformations is $m(x) = 2 \tan (3x + 4)$.

Frequently Asked Questions

Q: What is the parent tangent function?

A: The parent tangent function is given by $f(x) = \tan x$.

Q: What is the first transformation applied to the parent tangent function?

A: The first transformation applied to the parent tangent function is a horizontal compression by a factor of $\frac{1}{3}$.

Q: What is the second transformation applied to the parent tangent function?

A: The second transformation applied to the parent tangent function is a phase shift left by $\frac{4}{3}$.

Q: What is the third transformation applied to the parent tangent function?

A: The third transformation applied to the parent tangent function is a vertical stretch by a factor of $2$.

Q: What is the final function obtained after applying all the transformations?

$$

Q: What is the period of the final function $m(x) = 2 \tan (3x + 4)$?

A: The period of the final function $m(x) = 2 \tan (3x + 4)$ is $\frac{\pi}{3}$.

Q: What is the vertical asymptote of the final function $m(x) = 2 \tan (3x + 4)$?

A: The vertical asymptote of the final function $m(x) = 2 \tan (3x + 4)$ is $x = \frac{2\pi}{3} + k\frac{\pi}{3}$, where $k$ is an integer.

Q: How do I determine the period and vertical asymptote of a tangent function?

A: To determine the period and vertical asymptote of a tangent function, you need to identify the coefficient of $x$ inside the tangent function. The period is given by $\frac{\pi}{|a|}$, where $a$ is the coefficient of $x$. The vertical asymptote is given by $x = \frac{\pi}{2|a|} + k\frac{\pi}{|a|}$, where $k$ is an integer.

Q: Can I apply multiple transformations to a tangent function?

A: Yes, you can apply multiple transformations to a tangent function. Each transformation will modify the function in a specific way, and the final function will be a combination of all the transformations.

Q: How do I apply multiple transformations to a tangent function?

A: To apply multiple transformations to a tangent function, you need to follow the order of operations. First, apply the horizontal compression or stretch, then apply the phase shift, and finally apply the vertical stretch or compression.

Q: What is the difference between a horizontal compression and a horizontal stretch?

A: A horizontal compression is a transformation that makes the function narrower, while a horizontal stretch is a transformation that makes the function wider.

Q: What is the difference between a phase shift left and a phase shift right?

A: A phase shift left is a transformation that moves the function to the left, while a phase shift right is a transformation that moves the function to the right.

Q: Can I apply a phase shift to a function that has already been compressed or stretched?

A: Yes, you can apply a phase shift to a function that has already been compressed or stretched. The phase shift will modify the function in a specific way, and the final function will be a combination of all the transformations.

Q: How do I determine the coefficient of $x$ inside a tangent function?

A: To determine the coefficient of $x$ inside a tangent function, you need to look at the expression inside the tangent function. The coefficient of $x$ is the number that multiplies $x$.

Q: Can I apply a vertical stretch or compression to a function that has already been compressed or stretched?

A: Yes, you can apply a vertical stretch or compression to a function that has already been compressed or stretched. The vertical stretch or compression will modify the function in a specific way, and the final function will be a combination of all the transformations.

Q: How do I determine the vertical asymptote of a tangent function?

A: To determine the vertical asymptote of a tangent function, you need to identify the coefficient of $x$ inside the tangent function. The vertical asymptote is given by $x = \frac{\pi}{2|a|} + k\frac{\pi}{|a|}$, where $k$ is an integer.

Q: Can I apply multiple vertical stretches or compressions to a tangent function?

A: Yes, you can apply multiple vertical stretches or compressions to a tangent function. Each vertical stretch or compression will modify the function in a specific way, and the final function will be a combination of all the transformations.

Q: How do I apply multiple vertical stretches or compressions to a tangent function?

A: To apply multiple vertical stretches or compressions to a tangent function, you need to follow the order of operations. First, apply the vertical stretch or compression, then apply the horizontal compression or stretch, and finally apply the phase shift.

Q: What is the difference between a vertical stretch and a vertical compression?

A: A vertical stretch is a transformation that makes the function taller, while a vertical compression is a transformation that makes the function shorter.

Q: Can I apply a vertical stretch or compression to a function that has already been compressed or stretched?

A: Yes, you can apply a vertical stretch or compression to a function that has already been compressed or stretched. The vertical stretch or compression will modify the function in a specific way, and the final function will be a combination of all the transformations.