Select The Correct Statements In The Passage.Brian Wrote A Description Of The Transformations To The Parent Sine Function That Result In This Function:$\[ P(x) = -\frac{1}{4} \sin(x + \pi) - 2 \\]Which Statements In His Description Are True

Introduction

In mathematics, the sine function is a fundamental concept in trigonometry, and its transformations play a crucial role in understanding various mathematical concepts. The given function, $p(x) = -\frac{1}{4} \sin(x + \pi) - 2$, is a transformed version of the parent sine function. In this article, we will analyze the given function and determine which statements in Brian's description are true.

The Parent Sine Function

The parent sine function is given by $f(x) = \sin(x)$. This function has a period of $2\pi$, an amplitude of 1, and a phase shift of 0.

Transformations of the Sine Function

The given function, $p(x) = -\frac{1}{4} \sin(x + \pi) - 2$, is a transformed version of the parent sine function. The transformations include:

• Vertical Stretching: The coefficient of the sine function, $-\frac{1}{4}$, represents a vertical stretching of the parent sine function by a factor of $\frac{1}{4}$.
• Phase Shift: The term $x + \pi$ inside the sine function represents a phase shift of $\pi$ units to the left.
• Vertical Translation: The constant term, $-2$, represents a vertical translation of the function down by 2 units.

Analyzing the Statements

Let's analyze the statements in Brian's description and determine which ones are true.

Statement 1: The function has been vertically stretched by a factor of $\frac{1}{4}$.

• True: The coefficient of the sine function, $-\frac{1}{4}$, represents a vertical stretching of the parent sine function by a factor of $\frac{1}{4}$.
• Explanation: The vertical stretching factor is the absolute value of the coefficient of the sine function, which is $\frac{1}{4}$.

Statement 2: The function has been shifted $\pi$ units to the left.

• True: The term $x + \pi$ inside the sine function represents a phase shift of $\pi$ units to the left.
• Explanation: The phase shift is determined by the value inside the sine function, which is $x + \pi$. This means that the function has been shifted $\pi$ units to the left.

Statement 3: The function has been shifted down by 2 units.

• True: The constant term, $-2$, represents a vertical translation of the function down by 2 units.
• Explanation: The vertical translation is determined by the constant term inside the function, which is $-2$. This means that the function has been shifted down by 2 units.

Statement 4: The function has been vertically compressed by a factor of $\frac{1}{4}$.

• False: The coefficient of the sine function, $-\frac{1}{4}$, represents a vertical stretching of the parent sine function by a factor of $\frac{1}{4}$, not a vertical compression.
• Explanation: The vertical stretching factor is the absolute value of the coefficient of the sine function, which is $\frac{1}{4}$. This means that the function has been stretched, not compressed.

Statement 5: The function has been shifted $\pi$ units to the right.

• False: The term $x + \pi$ inside the sine function represents a phase shift of $\pi$ units to the left, not to the right.
• Explanation: The phase shift is determined by the value inside the sine function, which is $x + \pi$. This means that the function has been shifted $\pi$ units to the left, not to the right.

Conclusion

In conclusion, the statements in Brian's description that are true are:

• The function has been vertically stretched by a factor of $\frac{1}{4}$.
• The function has been shifted $\pi$ units to the left.
• The function has been shifted down by 2 units.

The statements that are false are:

• The function has been vertically compressed by a factor of $\frac{1}{4}$.
• The function has been shifted $\pi$ units to the right.

Q: What is the parent sine function?

A: The parent sine function is given by $f(x) = \sin(x)$. This function has a period of $2\pi$, an amplitude of 1, and a phase shift of 0.

Q: What are the transformations of the sine function?

A: The transformations of the sine function include:

• Vertical Stretching: A vertical stretching of the parent sine function by a factor of $\frac{1}{4}$.
• Phase Shift: A phase shift of $\pi$ units to the left.
• Vertical Translation: A vertical translation of the function down by 2 units.

Q: How do you determine the vertical stretching factor?

A: The vertical stretching factor is the absolute value of the coefficient of the sine function. In this case, the coefficient is $-\frac{1}{4}$, so the vertical stretching factor is $\frac{1}{4}$.

Q: How do you determine the phase shift?

A: The phase shift is determined by the value inside the sine function. In this case, the value is $x + \pi$, so the phase shift is $\pi$ units to the left.

Q: How do you determine the vertical translation?

A: The vertical translation is determined by the constant term inside the function. In this case, the constant term is $-2$, so the vertical translation is down by 2 units.

Q: What is the period of the transformed sine function?

A: The period of the transformed sine function is still $2\pi$, since the phase shift is $\pi$ units to the left, which does not change the period.

Q: What is the amplitude of the transformed sine function?

A: The amplitude of the transformed sine function is still 1, since the vertical stretching factor is $\frac{1}{4}$, which does not change the amplitude.

Q: How do you graph the transformed sine function?

A: To graph the transformed sine function, you can use the following steps:

1. Graph the parent sine function, $f(x) = \sin(x)$.
2. Apply the vertical stretching by a factor of $\frac{1}{4}$.
3. Apply the phase shift of $\pi$ units to the left.
4. Apply the vertical translation down by 2 units.

Q: What are some real-world applications of the transformations of the sine function?

A: The transformations of the sine function have many real-world applications, including:

• Modeling periodic phenomena: The sine function can be used to model periodic phenomena, such as the motion of a pendulum or the tides.
• Analyzing data: The sine function can be used to analyze data that exhibits periodic behavior, such as the stock market or the weather.
• Solving problems: The sine function can be used to solve problems that involve periodic motion, such as the motion of a spring or the vibration of a guitar string.

Conclusion

In conclusion, the transformations of the sine function are an important concept in mathematics that has many real-world applications. By understanding the vertical stretching, phase shift, and vertical translation of the sine function, you can model periodic phenomena, analyze data, and solve problems that involve periodic motion.