Enter The Correct Answer In The Box.Function { G $}$ Is The Result Of These Transformations On The Parent Sine Function:- Vertical Stretch By A Factor Of 3- Horizontal Shift Left { \frac{\pi}{2}$}$ Units- Vertical Shift Down 4

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Introduction

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In mathematics, the sine function is a fundamental concept in trigonometry, and it has numerous applications in various fields such as physics, engineering, and computer science. The sine function can be transformed in various ways to create new functions that can be used to model real-world phenomena. In this article, we will discuss the transformations of the sine function, specifically the vertical stretch, horizontal shift, and vertical shift.

Vertical Stretch

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A vertical stretch by a factor of 3 is a transformation that stretches the sine function vertically by a factor of 3. This means that the amplitude of the sine function is increased by a factor of 3. The new function is given by:

$$f(x) = 3 \sin(x)$$

The graph of the new function is a stretched version of the original sine function, with the amplitude increased by a factor of 3.

Horizontal Shift

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

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

The graph of the new function is a shifted version of the original sine function, with the graph shifted to the left by $\frac{\pi}{2}$ units.

Vertical Shift

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A vertical shift down 4 units is a transformation that shifts the sine function down by 4 units. This means that the graph of the sine function is shifted down by 4 units. The new function is given by:

$$f(x) = \sin(x) - 4$$

The graph of the new function is a shifted version of the original sine function, with the graph shifted down by 4 units.

Combining Transformations

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When combining transformations, the order in which they are applied is important. In general, the order of application is:

1. Horizontal shift
2. Vertical shift
3. Vertical stretch

This means that if we want to apply a horizontal shift, followed by a vertical shift, and then a vertical stretch, we would apply the transformations in the following order:

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

Example

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Let's consider an example to illustrate the transformations of the sine function. Suppose we want to find the function that results from 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.

Using the transformations discussed above, we can find the new function by applying the transformations in the correct order:

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

The graph of this function is a stretched, shifted, and shifted version of the original sine function.

Conclusion

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In conclusion, the transformations of the sine function are an important concept in mathematics, and they have numerous applications in various fields. By understanding the vertical stretch, horizontal shift, and vertical shift transformations, we can create new functions that can be used to model real-world phenomena. The order in which the transformations are applied is important, and we must apply the transformations in the correct order to obtain the desired function.

Key Takeaways

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• The vertical stretch transformation increases the amplitude of the sine function by a factor of 3.
• The horizontal shift transformation shifts the sine function left by $\frac{\pi}{2}$ units.
• The vertical shift transformation shifts the sine function down by 4 units.
• The order in which the transformations are applied is important, and we must apply the transformations in the correct order to obtain the desired function.

Further Reading

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For further reading on the transformations of the sine function, we recommend the following resources:

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

These resources provide a comprehensive introduction to the transformations of the sine function and their applications in various fields.

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Introduction

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In our previous article, we discussed the transformations of the sine function, including vertical stretch, horizontal shift, and vertical shift. In this article, we will answer some frequently asked questions about the transformations of the sine function.

Q: What is the effect of a vertical stretch on the sine function?

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A: A vertical stretch by a factor of 3 increases the amplitude of the sine function by a factor of 3. This means that the graph of the sine function is stretched vertically by a factor of 3.

Q: How does a horizontal shift affect the sine function?

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

Q: What is the effect of a vertical shift on the sine function?

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A: A vertical shift down 4 units shifts the graph of the sine function down by 4 units. This means that the graph of the sine function is shifted vertically down by 4 units.

Q: How do I combine multiple transformations of the sine function?

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A: When combining multiple transformations, the order in which they are applied is important. In general, the order of application is:

1. Horizontal shift
2. Vertical shift
3. Vertical stretch

For example, if we want to apply a horizontal shift left by $\frac{\pi}{2}$ units, followed by a vertical shift down 4 units, and then a vertical stretch by a factor of 3, we would apply the transformations in the following order:

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

Q: Can I apply multiple vertical shifts to the sine function?

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A: Yes, you can apply multiple vertical shifts to the sine function. For example, if we want to apply a vertical shift down 4 units, followed by a vertical shift down 2 units, we would apply the transformations in the following order:

$$f(x) = \sin(x) - 4 - 2$$

Q: Can I apply multiple horizontal shifts to the sine function?

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A: Yes, you can apply multiple horizontal shifts to the sine function. For example, if we want to apply a horizontal shift left by $\frac{\pi}{2}$ units, followed by a horizontal shift left by $\frac{\pi}{4}$ units, we would apply the transformations in the following order:

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

Q: How do I graph the transformed sine function?

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A: To graph the transformed sine function, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the graph of the function.

Q: Can I use the transformed sine function to model real-world phenomena?

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A: Yes, you can use the transformed sine function to model real-world phenomena. For example, you can use the transformed sine function to model the motion of a pendulum or the vibration of a spring.

Conclusion

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In conclusion, the transformations of the sine function are an important concept in mathematics, and they have numerous applications in various fields. By understanding the vertical stretch, horizontal shift, and vertical shift transformations, we can create new functions that can be used to model real-world phenomena.

Key Takeaways

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• The vertical stretch transformation increases the amplitude of the sine function by a factor of 3.
• The horizontal shift transformation shifts the sine function left by $\frac{\pi}{2}$ units.
• The vertical shift transformation shifts the sine function down by 4 units.
• The order in which the transformations are applied is important, and we must apply the transformations in the correct order to obtain the desired function.

Further Reading

---

For further reading on the transformations of the sine function, we recommend the following resources:

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

These resources provide a comprehensive introduction to the transformations of the sine function and their applications in various fields.