Which Set Of Transformations Is Needed To Graph F ( X ) = − 2 Sin ⁡ ( X ) + 3 F(x) = -2 \sin(x) + 3 F ( X ) = − 2 Sin ( X ) + 3 From The Parent Sine Function?A. Vertical Compression By A Factor Of 2, Vertical Translation 3 Units Up, Reflection Across The Y Y Y -axisB. Vertical Compression By

Understanding the Parent Sine Function

The parent sine function is a fundamental concept in trigonometry, represented by the equation $f(x) = \sin(x)$. This function has a standard period of $2\pi$ and an amplitude of 1. It oscillates between the values of -1 and 1, with its maximum value at $x = \frac{\pi}{2}$ and its minimum value at $x = \frac{3\pi}{2}$.

Transforming the Sine Function

When transforming the sine function, we need to consider the following parameters:

• Amplitude: The amplitude of the sine function determines the vertical stretch or compression of the graph. A positive amplitude indicates a vertical stretch, while a negative amplitude indicates a vertical compression.
• Period: The period of the sine function determines the horizontal stretch or compression of the graph. A period of $2\pi$ is the standard period for the sine function.
• Phase Shift: The phase shift of the sine function determines the horizontal translation of the graph. A phase shift of $\frac{\pi}{2}$ indicates a horizontal translation to the right, while a phase shift of $-\frac{\pi}{2}$ indicates a horizontal translation to the left.
• Vertical Translation: The vertical translation of the sine function determines the vertical shift of the graph. A vertical translation of $c$ indicates a shift of $c$ units up or down.

Transforming $f(x) = -2 \sin(x) + 3$

To graph the function $f(x) = -2 \sin(x) + 3$, we need to apply the following transformations to the parent sine function:

• Vertical Compression by a Factor of 2: The coefficient of the sine function, -2, indicates a vertical compression by a factor of 2. This means that the graph of the function will be compressed vertically by a factor of 2 compared to the parent sine function.
• Vertical Translation 3 Units Up: The constant term, 3, indicates a vertical translation of 3 units up. This means that the graph of the function will be shifted 3 units up compared to the parent sine function.

Reflection Across the $y$-Axis

The function $f(x) = -2 \sin(x) + 3$ also involves a reflection across the $y$-axis. This means that the graph of the function will be reflected across the $y$-axis, resulting in a mirror image of the parent sine function.

Conclusion

In conclusion, to graph the function $f(x) = -2 \sin(x) + 3$, we need to apply the following transformations to the parent sine function:

• Vertical compression by a factor of 2: This means that the graph of the function will be compressed vertically by a factor of 2 compared to the parent sine function.
• Vertical translation 3 units up: This means that the graph of the function will be shifted 3 units up compared to the parent sine function.
• Reflection across the $y$-axis: This means that the graph of the function will be reflected across the $y$-axis, resulting in a mirror image of the parent sine function.

By applying these transformations, we can graph the function $f(x) = -2 \sin(x) + 3$ and understand its behavior.

Example Use Cases

The function $f(x) = -2 \sin(x) + 3$ has several example use cases in various fields, including:

• Physics: The function can be used to model the motion of a pendulum or a spring-mass system.
• Engineering: The function can be used to model the behavior of electrical circuits or mechanical systems.
• Biology: The function can be used to model the growth of populations or the behavior of biological systems.

Conclusion

In conclusion, the function $f(x) = -2 \sin(x) + 3$ involves a vertical compression by a factor of 2, a vertical translation 3 units up, and a reflection across the $y$-axis. By applying these transformations to the parent sine function, we can graph the function and understand its behavior. The function has several example use cases in various fields, including physics, engineering, and biology.

References

• Boyer, C. B. (1968). A History of Mathematics. New York: Wiley.
• Krantz, S. G. (1997). A First Course in Mathematical Analysis. Cambridge University Press.
• Spivak, M. (1965). Calculus. New York: W. A. Benjamin.

Glossary

• Amplitude: The amplitude of a function determines the vertical stretch or compression of the graph.
• Period: The period of a function determines the horizontal stretch or compression of the graph.
• Phase Shift: The phase shift of a function determines the horizontal translation of the graph.
• Vertical Translation: The vertical translation of a function determines the vertical shift of the graph.

Further Reading

For further reading on the topic of transforming sine functions, we recommend the following resources:

• Boyer, C. B. (1968). A History of Mathematics. New York: Wiley.
• Krantz, S. G. (1997). A First Course in Mathematical Analysis. Cambridge University Press.
• Spivak, M. (1965). Calculus. New York: W. A. Benjamin.

FAQs

Q: What is the parent sine function? A: The parent sine function is a fundamental concept in trigonometry, represented by the equation $f(x) = \sin(x)$.

Q: What are the parameters of the sine function? A: The parameters of the sine function include amplitude, period, phase shift, and vertical translation.

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Q: What is the parent sine function?

A: The parent sine function is a fundamental concept in trigonometry, represented by the equation $f(x) = \sin(x)$. This function has a standard period of $2\pi$ and an amplitude of 1. It oscillates between the values of -1 and 1, with its maximum value at $x = \frac{\pi}{2}$ and its minimum value at $x = \frac{3\pi}{2}$.

Q: What are the parameters of the sine function?

A: The parameters of the sine function include:

• Amplitude: The amplitude of the sine function determines the vertical stretch or compression of the graph. A positive amplitude indicates a vertical stretch, while a negative amplitude indicates a vertical compression.
• Period: The period of the sine function determines the horizontal stretch or compression of the graph. A period of $2\pi$ is the standard period for the sine function.
• Phase Shift: The phase shift of the sine function determines the horizontal translation of the graph. A phase shift of $\frac{\pi}{2}$ indicates a horizontal translation to the right, while a phase shift of $-\frac{\pi}{2}$ indicates a horizontal translation to the left.
• Vertical Translation: The vertical translation of the sine function determines the vertical shift of the graph. A vertical translation of $c$ indicates a shift of $c$ units up or down.

Q: How do I graph the function $f(x) = -2 \sin(x) + 3$?

A: To graph the function $f(x) = -2 \sin(x) + 3$, you need to apply the following transformations to the parent sine function:

• Vertical Compression by a Factor of 2: The coefficient of the sine function, -2, indicates a vertical compression by a factor of 2. This means that the graph of the function will be compressed vertically by a factor of 2 compared to the parent sine function.
• Vertical Translation 3 Units Up: The constant term, 3, indicates a vertical translation of 3 units up. This means that the graph of the function will be shifted 3 units up compared to the parent sine function.
• Reflection Across the $y$-Axis: The negative sign in front of the sine function indicates a reflection across the $y$-axis. This means that the graph of the function will be reflected across the $y$-axis, resulting in a mirror image of the parent sine function.

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

A: Some common transformations of the sine function include:

• Vertical Stretch or Compression: A vertical stretch or compression of the sine function can be achieved by multiplying the sine function by a positive or negative constant.
• Horizontal Stretch or Compression: A horizontal stretch or compression of the sine function can be achieved by replacing $x$ with $ax$ or $x/a$.
• Phase Shift: A phase shift of the sine function can be achieved by replacing $x$ with $x + c$ or $x - c$.
• Vertical Translation: A vertical translation of the sine function can be achieved by adding a constant to the sine function.

Q: How do I determine the amplitude of a sine function?

A: The amplitude of a sine function can be determined by looking at the coefficient of the sine function. If the coefficient is positive, the amplitude is the absolute value of the coefficient. If the coefficient is negative, the amplitude is the absolute value of the coefficient multiplied by -1.

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

A: The period of a sine function can be determined by looking at the coefficient of $x$ in the sine function. If the coefficient is 1, the period is $2\pi$. If the coefficient is $a$, the period is $2\pi/a$.

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

A: The phase shift of a sine function can be determined by looking at the constant term in the sine function. If the constant term is $c$, the phase shift is $c$.

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

A: The vertical translation of a sine function can be determined by looking at the constant term in the sine function. If the constant term is $c$, the vertical translation is $c$.

Q: What are some real-world applications of sine functions?

A: Sine functions have many real-world applications, including:

• Physics: Sine functions are used to model the motion of objects, such as pendulums and springs.
• Engineering: Sine functions are used to model the behavior of electrical circuits and mechanical systems.
• Biology: Sine functions are used to model the growth of populations and the behavior of biological systems.

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

A: To graph a sine function with a phase shift, you need to apply the following transformations to the parent sine function:

• Horizontal Translation: Replace $x$ with $x + c$ or $x - c$ to achieve a horizontal translation.
• Vertical Translation: Add a constant to the sine function to achieve a vertical translation.

Q: How do I graph a sine function with a vertical stretch or compression?

A: To graph a sine function with a vertical stretch or compression, you need to apply the following transformations to the parent sine function:

• Vertical Stretch or Compression: Multiply the sine function by a positive or negative constant to achieve a vertical stretch or compression.

Q: How do I graph a sine function with a horizontal stretch or compression?

A: To graph a sine function with a horizontal stretch or compression, you need to apply the following transformations to the parent sine function:

• Horizontal Stretch or Compression: Replace $x$ with $ax$ or $x/a$ to achieve a horizontal stretch or compression.