Understanding the Parent Sine Function

Oct 13, 2025 by windailys.com 130 views

Understanding the Parent Sine Function

The parent sine function is a fundamental concept in mathematics, particularly in trigonometry. It is defined as:

f(x) = \sin(x)

This function has a period of $2\pi$ , meaning it repeats every $2\pi$ units. The graph of the sine function is a wave that oscillates between $-1$ and $1$ .

Vertical Stretch by a Factor of 3

The first transformation applied to the parent sine function is a vertical stretch by a factor of 3. This means that the amplitude of the function is increased by a factor of 3. The new function is:

f(x) = 3\sin(x)

The graph of this function will have a greater amplitude than the parent sine function, with a maximum value of 3 and a minimum value of -3.

Horizontal Shift Left by $\frac{\pi}{2}$ Units

The next transformation applied to the function is a horizontal shift left by $\frac{\pi}{2}$ units. This means that the graph of the function is shifted to the left by $\frac{\pi}{2}$ units. The new function is:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right)

The graph of this function will have the same amplitude as the previous function, but it will be shifted to the left by $\frac{\pi}{2}$ units.

Vertical Shift Down 4 Units

The final transformation applied to the function is a vertical shift down 4 units. This means that the graph of the function is shifted down by 4 units. The new function is:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right) - 4

The graph of this function will have the same amplitude and period as the previous function, but it will be shifted down by 4 units.

The Final Function

The final function is the result of all the transformations applied to the parent sine function. It is:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right) - 4

This function has a period of $2\pi$ , an amplitude of 3, and a vertical shift of -4 units.

Graph of the Final Function

The graph of the final function is a wave that oscillates between -7 and 1. It has a period of $2\pi$ , an amplitude of 3, and a vertical shift of -4 units.

Key Features of the Final Function

Period: The period of the final function is $2\pi$ , which means it repeats every $2\pi$ units.
Amplitude: The amplitude of the final function is 3, which means it oscillates between -3 and 3.
Vertical Shift: The final function has a vertical shift of -4 units, which means it is shifted down by 4 units.

Conclusion

In conclusion, the final function is the result of applying a vertical stretch by a factor of 3, a horizontal shift left by $\frac{\pi}{2}$ units, and a vertical shift down 4 units to the parent sine function. The graph of the final function is a wave that oscillates between -7 and 1, with a period of $2\pi$ , an amplitude of 3, and a vertical shift of -4 units.

Example Problems

Find the value of the final function at $x = 0$ .
Find the value of the final function at $x = \pi$ .
Find the period of the final function.
Find the amplitude of the final function.
Find the vertical shift of the final function.

Solutions

The value of the final function at $x = 0$ is:

f(0) = 3\sin\left(0 + \frac{\pi}{2}\right) - 4

f(0) = 3\sin\left(\frac{\pi}{2}\right) - 4

f(0) = 3(1) - 4

f(0) = -1

The value of the final function at $x = \pi$ is:

f(\pi) = 3\sin\left(\pi + \frac{\pi}{2}\right) - 4

f(\pi) = 3\sin\left(\frac{3\pi}{2}\right) - 4

f(\pi) = 3(-1) - 4

f(\pi) = -7

The period of the final function is $2\pi$ .
The amplitude of the final function is 3.
The vertical shift of the final function is -4 units.

Practice Problems

Find the value of the final function at $x = \frac{\pi}{2}$ .
Find the value of the final function at $x = \frac{3\pi}{2}$ .
Find the period of the function $f(x) = 2\sin(x)$ .
Find the amplitude of the function $f(x) = 2\sin(x)$ .
Find the vertical shift of the function $f(x) = 2\sin(x)$ .

Solutions

The value of the final function at $x = \frac{\pi}{2}$ is:

f\left(\frac{\pi}{2}\right) = 3\sin\left(\frac{\pi}{2} + \frac{\pi}{2}\right) - 4

f\left(\frac{\pi}{2}\right) = 3\sin(\pi) - 4

f\left(\frac{\pi}{2}\right) = 3(0) - 4

f\left(\frac{\pi}{2}\right) = -4

The value of the final function at $x = \frac{3\pi}{2}$ is:

f\left(\frac{3\pi}{2}\right) = 3\sin\left(\frac{3\pi}{2} + \frac{\pi}{2}\right) - 4

f\left(\frac{3\pi}{2}\right) = 3\sin(2\pi) - 4

f\left(\frac{3\pi}{2}\right) = 3(0) - 4

f\left(\frac{3\pi}{2}\right) = -4

The period of the function $f(x) = 2\sin(x)$ is $2\pi$ .
The amplitude of the function $f(x) = 2\sin(x)$ is 2.
The vertical shift of the function $f(x) = 2\sin(x)$ is 0 units.

Conclusion

$In conclusion, the final function is the result of applying a vertical stretch by a factor of 3, a horizontal shift left by $\frac{\pi}{2}$ units, and a vertical shift down 4 units to the parent sine function. The graph of the final function is a wave that oscillates between -7 and 1, with a period of $2\pi$, an amplitude of 3, and a vertical shift of -4 units.<br/> **Q&A: Transformations of the Sine Function**$

Q: What is the parent sine function?

A: The parent sine function is a fundamental concept in mathematics, particularly in trigonometry. It is defined as:

f(x) = \sin(x)

This function has a period of $2\pi$ , meaning it repeats every $2\pi$ units. The graph of the sine function is a wave that oscillates between $-1$ and $1$ .

Q: What is a vertical stretch by a factor of 3?

A: A vertical stretch by a factor of 3 means that the amplitude of the function is increased by a factor of 3. The new function is:

f(x) = 3\sin(x)

The graph of this function will have a greater amplitude than the parent sine function, with a maximum value of 3 and a minimum value of -3.

Q: What is a horizontal shift left by $\frac{\pi}{2}$ units?

A: A horizontal shift left by $\frac{\pi}{2}$ units means that the graph of the function is shifted to the left by $\frac{\pi}{2}$ units. The new function is:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right)

The graph of this function will have the same amplitude as the previous function, but it will be shifted to the left by $\frac{\pi}{2}$ units.

Q: What is a vertical shift down 4 units?

A: A vertical shift down 4 units means that the graph of the function is shifted down by 4 units. The new function is:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right) - 4

The graph of this function will have the same amplitude and period as the previous function, but it will be shifted down by 4 units.

Q: What is the final function?

A: The final function is the result of all the transformations applied to the parent sine function. It is:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right) - 4

This function has a period of $2\pi$ , an amplitude of 3, and a vertical shift of -4 units.

Q: What is the graph of the final function?

A: The graph of the final function is a wave that oscillates between -7 and 1. It has a period of $2\pi$ , an amplitude of 3, and a vertical shift of -4 units.

Q: What are the key features of the final function?

A: The key features of the final function are:

Period: The period of the final function is $2\pi$ , which means it repeats every $2\pi$ units.
Amplitude: The amplitude of the final function is 3, which means it oscillates between -3 and 3.
Vertical Shift: The final function has a vertical shift of -4 units, which means it is shifted down by 4 units.

Q: How do I find the value of the final function at a given point?

A: To find the value of the final function at a given point, you can plug the value of x into the equation:

f(x) = 3\sin\left(x + \frac{\pi}{2}\right) - 4

For example, to find the value of the final function at x = 0, you would plug x = 0 into the equation:

f(0) = 3\sin\left(0 + \frac{\pi}{2}\right) - 4

f(0) = 3\sin\left(\frac{\pi}{2}\right) - 4

f(0) = 3(1) - 4

f(0) = -1

Q: How do I find the period of the final function?

A: The period of the final function is $2\pi$ , which means it repeats every $2\pi$ units.

Q: How do I find the amplitude of the final function?

A: The amplitude of the final function is 3, which means it oscillates between -3 and 3.

Q: How do I find the vertical shift of the final function?

A: The final function has a vertical shift of -4 units, which means it is shifted down by 4 units.

Q: What are some examples of transformations of the sine function?

A: Some examples of transformations of the sine function include:

Vertical stretch by a factor of 2: $f(x) = 2\sin(x)$
Horizontal shift left by $\pi$ units: $f(x) = \sin(x + \pi)$
Vertical shift down 2 units: $f(x) = \sin(x) - 2$

Q: What are some practice problems for transformations of the sine function?

A: Some practice problems for transformations of the sine function include:

Find the value of the function $f(x) = 3\sin(x + \frac{\pi}{2}) - 4$ at x = 0.
Find the period of the function $f(x) = 2\sin(x)$ .
Find the amplitude of the function $f(x) = 2\sin(x)$ .
Find the vertical shift of the function $f(x) = 2\sin(x)$ .

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

A: Some real-world applications of transformations of the sine function include:

Modeling population growth: The sine function can be used to model population growth, where the amplitude represents the maximum population and the period represents the time it takes for the population to reach its maximum.
Modeling sound waves: The sine function can be used to model sound waves, where the amplitude represents the loudness of the sound and the period represents the frequency of the sound.
Modeling electrical signals: The sine function can be used to model electrical signals, where the amplitude represents the voltage of the signal and the period represents the frequency of the signal.