Understanding Phase Shifts

Oct 13, 2025 by windailys.com 32 views

Introduction

In trigonometry, phase shifts are used to describe the horizontal movement of a function's graph. A phase shift to the right indicates that the graph is shifted to the right by a certain number of units. In this article, we will explore which function has a phase shift of $\frac{\pi}{2}$ to the right.

Understanding Phase Shifts

A phase shift is a horizontal movement of a function's graph. It is represented by the equation $y = a \sin (bx + c)$ , where $a$ is the amplitude, $b$ is the frequency, and $c$ is the phase shift. The phase shift is given by the value of $c$ .

Phase Shift to the Right

A phase shift to the right is represented by a positive value of $c$ . This means that the graph is shifted to the right by the value of $c$ . In this case, we are looking for a function with a phase shift of $\frac{\pi}{2}$ to the right.

Analyzing the Options

Let's analyze each option to determine which one has a phase shift of $\frac{\pi}{2}$ to the right.

Option A: $y = 2 \sin (2x - \pi)$

In this option, the phase shift is given by the value of $- \pi$ . This is a negative value, which means that the graph is shifted to the left, not to the right. Therefore, this option is not correct.

Option B: $y = 2 \sin (x - \pi)$

Option C: $y = 2 \sin \left(x + \frac{\pi}{2}\right)$

In this option, the phase shift is given by the value of $\frac{\pi}{2}$ . This is a positive value, which means that the graph is shifted to the right. Therefore, this option is correct.

Option D: $y = 2 \sin \left(\frac{1}{2}x + \pi\right)$

In this option, the phase shift is given by the value of $\pi$ . This is a positive value, but it is not equal to $\frac{\pi}{2}$ . Therefore, this option is not correct.

Conclusion

In conclusion, the function with a phase shift of $\frac{\pi}{2}$ to the right is $y = 2 \sin \left(x + \frac{\pi}{2}\right)$ . This function has a positive phase shift, which means that the graph is shifted to the right.

Understanding the Graph

To understand the graph of the function $y = 2 \sin \left(x + \frac{\pi}{2}\right)$ , we need to analyze its components.

The amplitude of the function is 2, which means that the graph will oscillate between -2 and 2.
The frequency of the function is 1, which means that the graph will complete one cycle in 2π units.
The phase shift of the function is $\frac{\pi}{2}$ , which means that the graph will be shifted to the right by $\frac{\pi}{2}$ units.

Graphing the Function

To graph the function $y = 2 \sin \left(x + \frac{\pi}{2}\right)$ , we need to follow these steps:

Start by graphing the basic sine function, $y = \sin x$ .
Shift the graph to the right by $\frac{\pi}{2}$ units.
Stretch the graph vertically by a factor of 2.

Example Problems

Here are some example problems to help you practice graphing functions with phase shifts.

Example 1

Graph the function $y = 2 \sin \left(x + \frac{\pi}{2}\right)$ .

Solution

To graph the function, we need to follow the steps outlined above.

Start by graphing the basic sine function, $y = \sin x$ .
Shift the graph to the right by $\frac{\pi}{2}$ units.
Stretch the graph vertically by a factor of 2.

The final graph will be a sine curve with an amplitude of 2 and a phase shift of $\frac{\pi}{2}$ to the right.

Example 2

Graph the function $y = 2 \sin \left(x - \frac{\pi}{2}\right)$ .

Solution

To graph the function, we need to follow the steps outlined above.

Start by graphing the basic sine function, $y = \sin x$ .
Shift the graph to the left by $\frac{\pi}{2}$ units.
Stretch the graph vertically by a factor of 2.

The final graph will be a sine curve with an amplitude of 2 and a phase shift of $\frac{\pi}{2}$ to the left.

Practice Problems

Here are some practice problems to help you practice graphing functions with phase shifts.

Problem 1

Graph the function $y = 2 \sin \left(x + \pi\right)$ .

Problem 2

Graph the function $y = 2 \sin \left(x - \pi\right)$ .

Problem 3

Graph the function $y = 2 \sin \left(x + \frac{\pi}{4}\right)$ .

Problem 4

Graph the function $y = 2 \sin \left(x - \frac{\pi}{4}\right)$ .

Conclusion

$In conclusion, the function with a phase shift of $\frac{\pi}{2}$ to the right is $y = 2 \sin \left(x + \frac{\pi}{2}\right)$. This function has a positive phase shift, which means that the graph is shifted to the right. We also discussed how to graph functions with phase shifts and provided some example problems to help you practice.<br/> **Q&A: Functions with Phase Shifts**$

Frequently Asked Questions

In this article, we will answer some frequently asked questions about functions with phase shifts.

Q: What is a phase shift?

A: A phase shift is a horizontal movement of a function's graph. It is represented by the equation $y = a \sin (bx + c)$ , where $a$ is the amplitude, $b$ is the frequency, and $c$ is the phase shift.

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

A: A phase shift to the right is represented by a positive value of $c$ , while a phase shift to the left is represented by a negative value of $c$ .

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 value of $c$ in the equation $y = a \sin (bx + c)$ . If $c$ is positive, the graph is shifted to the right. If $c$ is negative, the graph is shifted to the left.

Q: Can a function have a phase shift of zero?

A: Yes, a function can have a phase shift of zero. This means that the graph is not shifted to the right or left.

Q: How do I graph a function with a phase shift?

A: To graph a function with a phase shift, you need to follow these steps:

Start by graphing the basic sine function, $y = \sin x$ .
Shift the graph to the right or left by the value of $c$ .
Stretch the graph vertically by a factor of $a$ .

Q: Can a function have a phase shift of $\frac{\pi}{2}$ ?

A: Yes, a function can have a phase shift of $\frac{\pi}{2}$ . This means that the graph is shifted to the right by $\frac{\pi}{2}$ units.

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

A: To determine the amplitude of a function with a phase shift, you need to look at the value of $a$ in the equation $y = a \sin (bx + c)$ . The amplitude is the maximum value of the function.

Q: Can a function have a phase shift and a negative amplitude?

A: Yes, a function can have a phase shift and a negative amplitude. This means that the graph is shifted to the right or left and the function oscillates between negative and positive values.

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

A: To determine the frequency of a function with a phase shift, you need to look at the value of $b$ in the equation $y = a \sin (bx + c)$ . The frequency is the number of cycles the function completes in $2\pi$ units.

Q: Can a function have a phase shift and a frequency of zero?

A: No, a function cannot have a phase shift and a frequency of zero. The frequency is always a positive value.

Q: How do I graph a function with a phase shift and a negative amplitude?

A: To graph a function with a phase shift and a negative amplitude, you need to follow these steps:

Start by graphing the basic sine function, $y = \sin x$ .
Shift the graph to the right or left by the value of $c$ .
Stretch the graph vertically by a factor of $-a$ .

Q: Can a function have a phase shift and a frequency of $\frac{1}{2}$ ?

A: Yes, a function can have a phase shift and a frequency of $\frac{1}{2}$ . This means that the graph is shifted to the right or left and the function completes one cycle in $4\pi$ units.

Conclusion

In conclusion, we have answered some frequently asked questions about functions with phase shifts. We have discussed how to determine the phase shift, amplitude, and frequency of a function, and how to graph a function with a phase shift. We have also provided some example problems to help you practice.