Suppose You Want To Transform The Graph Of The Function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ Into The Graph Of The Function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$. Which Transformations Should You Perform?A. Reflect The

Feb 25, 2025 by ADMIN 233 views

Introduction

Transforming graphs of trigonometric functions is a crucial concept in mathematics, particularly in calculus and algebra. It involves understanding how different transformations affect the graph of a function. In this article, we will explore how to transform the graph of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ into the graph of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ .

Understanding the Original Function

The original function is $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ . This function involves a tangent function with a phase shift of $\frac{\pi}{4}$ and a vertical shift of $-1$ . The tangent function has a period of $\pi$ , which means that the graph of the function repeats every $\pi$ units.

Understanding the Target Function

The target function is $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ . This function involves a negative tangent function with a phase shift of $\frac{\pi}{2}$ and a vertical shift of $1$ . The negative sign in front of the tangent function indicates that the graph of the function will be reflected across the x-axis.

Determining the Transformations

To transform the graph of the original function into the graph of the target function, we need to perform the following transformations:

Phase Shift

The original function has a phase shift of $\frac{\pi}{4}$ , while the target function has a phase shift of $\frac{\pi}{2}$ . To achieve the phase shift of $\frac{\pi}{2}$ , we need to shift the graph of the original function to the left by $\frac{\pi}{4}$ units.

Reflection

The original function is a tangent function, while the target function is a negative tangent function. To achieve the negative sign, we need to reflect the graph of the original function across the x-axis.

Vertical Shift

The original function has a vertical shift of $-1$ , while the target function has a vertical shift of $1$ . To achieve the vertical shift of $1$ , we need to shift the graph of the original function up by $2$ units.

Period

The original function has a period of $\pi$ , while the target function has a period of $\pi$ . The period of the function remains the same.

Performing the Transformations

To perform the transformations, we need to apply the following steps:

Phase Shift: Shift the graph of the original function to the left by $\frac{\pi}{4}$ units.
Reflection: Reflect the graph of the original function across the x-axis.
Vertical Shift: Shift the graph of the original function up by $2$ units.

Conclusion

Transforming graphs of trigonometric functions is a crucial concept in mathematics. By understanding how different transformations affect the graph of a function, we can transform the graph of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ into the graph of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ . The transformations involved are a phase shift, reflection, and vertical shift.

Example

Suppose we want to transform the graph of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ into the graph of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ . We can perform the following steps:

Phase Shift: Shift the graph of the original function to the left by $\frac{\pi}{4}$ units.
Reflection: Reflect the graph of the original function across the x-axis.
Vertical Shift: Shift the graph of the original function up by $2$ units.

By performing these transformations, we can transform the graph of the original function into the graph of the target function.

Applications

Transforming graphs of trigonometric functions has numerous applications in mathematics, physics, and engineering. Some of the applications include:

Calculus: Transforming graphs of trigonometric functions is essential in calculus, particularly in the study of limits, derivatives, and integrals.
Physics: Transforming graphs of trigonometric functions is crucial in physics, particularly in the study of wave motion, vibrations, and oscillations.
Engineering: Transforming graphs of trigonometric functions is essential in engineering, particularly in the design of electronic circuits, filters, and amplifiers.

Final Thoughts

Introduction

Transforming graphs of trigonometric functions is a crucial concept in mathematics, particularly in calculus and algebra. In our previous article, we explored how to transform the graph of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ into the graph of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ . In this article, we will answer some frequently asked questions about transforming graphs of trigonometric functions.

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

A: A phase shift is a horizontal shift of the graph of a function, while a vertical shift is a vertical shift of the graph of a function. In the case of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ , the phase shift is $\frac{\pi}{4}$ , while the vertical shift is $-1$ .

Q: How do I determine the phase shift of a function?

A: To determine the phase shift of a function, you need to look at the argument of the trigonometric function. In the case of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ , the argument is $x+\frac{\pi}{4}$ , which means that the phase shift is $\frac{\pi}{4}$ .

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

A: To determine the vertical shift of a function, you need to look at the constant term outside the trigonometric function. In the case of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ , the constant term is $-1$ , which means that the vertical shift is $-1$ .

Q: What is the difference between a reflection and a rotation?

A: A reflection is a flip of the graph of a function across a line, while a rotation is a rotation of the graph of a function around a point. In the case of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ , the reflection is across the x-axis, while the rotation is around the point $(0,1)$ .

Q: How do I determine the reflection of a function?

A: To determine the reflection of a function, you need to look at the sign of the trigonometric function. In the case of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ , the sign is negative, which means that the reflection is across the x-axis.

Q: How do I determine the rotation of a function?

A: To determine the rotation of a function, you need to look at the argument of the trigonometric function. In the case of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ , the argument is $x+\frac{\pi}{2}$ , which means that the rotation is around the point $(0,1)$ .

Q: What is the difference between a period and a frequency?

A: A period is the length of time it takes for a function to complete one cycle, while a frequency is the number of cycles per unit of time. In the case of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ , the period is $\pi$ , while the frequency is $\frac{1}{\pi}$ .

Q: How do I determine the period of a function?

A: To determine the period of a function, you need to look at the argument of the trigonometric function. In the case of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ , the argument is $x+\frac{\pi}{4}$ , which means that the period is $\pi$ .

Q: How do I determine the frequency of a function?

A: To determine the frequency of a function, you need to look at the period of the function. In the case of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ , the period is $\pi$ , which means that the frequency is $\frac{1}{\pi}$ .

Conclusion

Transforming graphs of trigonometric functions is a crucial concept in mathematics, particularly in calculus and algebra. By understanding how different transformations affect the graph of a function, we can transform the graph of the function $y = \tan \left(x+\frac{\pi}{4}\right) - 1$ into the graph of the function $y = -\tan \left(x+\frac{\pi}{2}\right) + 1$ . The transformations involved are a phase shift, reflection, and vertical shift. We hope that this Q&A article has helped to clarify any questions you may have had about transforming graphs of trigonometric functions.